Calculus Examples

$\frac{x^{3}}{x^{2} - 4}$

Step 1

Write $\frac{x^{3}}{x^{2} - 4}$ as a function.

$f (x) = \frac{x^{3}}{x^{2} - 4}$

Step 2

Find the $x$ values where the second derivative is equal to $0$ .

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Step 2.1

Find the second derivative.

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Step 2.1.1

Find the first derivative.

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Step 2.1.1.1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x^{3}$ and $g (x) = x^{2} - 4$ .

$\frac{(x^{2} - 4) \frac{d}{d x} [x^{3}] - x^{3} \frac{d}{d x} [x^{2} - 4]}{{(x^{2} - 4)}^{2}}$

Step 2.1.1.2

Differentiate.

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Step 2.1.1.2.1

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$\frac{(x^{2} - 4) (3 x^{2}) - x^{3} \frac{d}{d x} [x^{2} - 4]}{{(x^{2} - 4)}^{2}}$

Step 2.1.1.2.2

Move $3$ to the left of $x^{2} - 4$ .

$\frac{3 \cdot (x^{2} - 4) x^{2} - x^{3} \frac{d}{d x} [x^{2} - 4]}{{(x^{2} - 4)}^{2}}$

Step 2.1.1.2.3

By the Sum Rule, the derivative of $x^{2} - 4$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4]$ .

$\frac{3 (x^{2} - 4) x^{2} - x^{3} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4])}{{(x^{2} - 4)}^{2}}$

Step 2.1.1.2.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$\frac{3 (x^{2} - 4) x^{2} - x^{3} (2 x + \frac{d}{d x} [- 4])}{{(x^{2} - 4)}^{2}}$

Step 2.1.1.2.5

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

$\frac{3 (x^{2} - 4) x^{2} - x^{3} (2 x + 0)}{{(x^{2} - 4)}^{2}}$

Step 2.1.1.2.6

Simplify the expression.

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Step 2.1.1.2.6.1

Add $2 x$ and $0$ .

$\frac{3 (x^{2} - 4) x^{2} - x^{3} (2 x)}{{(x^{2} - 4)}^{2}}$

Step 2.1.1.2.6.2

Multiply $2$ by $- 1$ .

$\frac{3 (x^{2} - 4) x^{2} - 2 x^{3} x}{{(x^{2} - 4)}^{2}}$

Step 2.1.1.3

Raise $x$ to the power of $1$ .

$\frac{3 (x^{2} - 4) x^{2} - 2 (x^{1} x^{3})}{{(x^{2} - 4)}^{2}}$

Step 2.1.1.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{3 (x^{2} - 4) x^{2} - 2 x^{1 + 3}}{{(x^{2} - 4)}^{2}}$

Step 2.1.1.5

Add $1$ and $3$ .

$\frac{3 (x^{2} - 4) x^{2} - 2 x^{4}}{{(x^{2} - 4)}^{2}}$

Step 2.1.1.6

Simplify.

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Step 2.1.1.6.1

Apply the distributive property.

$\frac{(3 x^{2} + 3 \cdot - 4) x^{2} - 2 x^{4}}{{(x^{2} - 4)}^{2}}$

Step 2.1.1.6.2

Apply the distributive property.

$\frac{3 x^{2} x^{2} + 3 \cdot - 4 x^{2} - 2 x^{4}}{{(x^{2} - 4)}^{2}}$

Step 2.1.1.6.3

Simplify the numerator.

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Step 2.1.1.6.3.1

Simplify each term.

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Step 2.1.1.6.3.1.1

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 2.1.1.6.3.1.1.1

Move $x^{2}$ .

$\frac{3 (x^{2} x^{2}) + 3 \cdot - 4 x^{2} - 2 x^{4}}{{(x^{2} - 4)}^{2}}$

Step 2.1.1.6.3.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{3 x^{2 + 2} + 3 \cdot - 4 x^{2} - 2 x^{4}}{{(x^{2} - 4)}^{2}}$

Step 2.1.1.6.3.1.1.3

Add $2$ and $2$ .

$\frac{3 x^{4} + 3 \cdot - 4 x^{2} - 2 x^{4}}{{(x^{2} - 4)}^{2}}$

Step 2.1.1.6.3.1.2

Multiply $3$ by $- 4$ .

$\frac{3 x^{4} - 12 x^{2} - 2 x^{4}}{{(x^{2} - 4)}^{2}}$

Step 2.1.1.6.3.2

Subtract $2 x^{4}$ from $3 x^{4}$ .

$\frac{x^{4} - 12 x^{2}}{{(x^{2} - 4)}^{2}}$

Step 2.1.1.6.4

Factor $x^{2}$ out of $x^{4} - 12 x^{2}$ .

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Step 2.1.1.6.4.1

Factor $x^{2}$ out of $x^{4}$ .

$\frac{x^{2} x^{2} - 12 x^{2}}{{(x^{2} - 4)}^{2}}$

Step 2.1.1.6.4.2

Factor $x^{2}$ out of $- 12 x^{2}$ .

$\frac{x^{2} x^{2} + x^{2} \cdot - 12}{{(x^{2} - 4)}^{2}}$

Step 2.1.1.6.4.3

Factor $x^{2}$ out of $x^{2} x^{2} + x^{2} \cdot - 12$ .

$\frac{x^{2} (x^{2} - 12)}{{(x^{2} - 4)}^{2}}$

Step 2.1.1.6.5

Simplify the denominator.

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Step 2.1.1.6.5.1

Rewrite $4$ as $2^{2}$ .

$\frac{x^{2} (x^{2} - 12)}{{(x^{2} - 2^{2})}^{2}}$

Step 2.1.1.6.5.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 2$ .

$\frac{x^{2} (x^{2} - 12)}{{((x + 2) (x - 2))}^{2}}$

Step 2.1.1.6.5.3

Apply the product rule to $(x + 2) (x - 2)$ .

$f' (x) = \frac{x^{2} (x^{2} - 12)}{{(x + 2)}^{2} {(x - 2)}^{2}}$

Step 2.1.2

Find the second derivative.

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Step 2.1.2.1

$\frac{{(x + 2)}^{2} {(x - 2)}^{2} \frac{d}{d x} [x^{2} (x^{2} - 12)] - x^{2} (x^{2} - 12) \frac{d}{d x} [{(x + 2)}^{2} {(x - 2)}^{2}]}{{({(x + 2)}^{2} {(x - 2)}^{2})}^{2}}$

Step 2.1.2.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{2}$ and $g (x) = x^{2} - 12$ .

$\frac{{(x + 2)}^{2} {(x - 2)}^{2} (x^{2} \frac{d}{d x} [x^{2} - 12] + (x^{2} - 12) \frac{d}{d x} [x^{2}]) - x^{2} (x^{2} - 12) \frac{d}{d x} [{(x + 2)}^{2} {(x - 2)}^{2}]}{{({(x + 2)}^{2} {(x - 2)}^{2})}^{2}}$

Step 2.1.2.3

Differentiate.

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Step 2.1.2.3.1

By the Sum Rule, the derivative of $x^{2} - 12$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 12]$ .

$\frac{{(x + 2)}^{2} {(x - 2)}^{2} (x^{2} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 12]) + (x^{2} - 12) \frac{d}{d x} [x^{2}]) - x^{2} (x^{2} - 12) \frac{d}{d x} [{(x + 2)}^{2} {(x - 2)}^{2}]}{{({(x + 2)}^{2} {(x - 2)}^{2})}^{2}}$

Step 2.1.2.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$\frac{{(x + 2)}^{2} {(x - 2)}^{2} (x^{2} (2 x + \frac{d}{d x} [- 12]) + (x^{2} - 12) \frac{d}{d x} [x^{2}]) - x^{2} (x^{2} - 12) \frac{d}{d x} [{(x + 2)}^{2} {(x - 2)}^{2}]}{{({(x + 2)}^{2} {(x - 2)}^{2})}^{2}}$

Step 2.1.2.3.3

Since $- 12$ is constant with respect to $x$ , the derivative of $- 12$ with respect to $x$ is $0$ .

$\frac{{(x + 2)}^{2} {(x - 2)}^{2} (x^{2} (2 x + 0) + (x^{2} - 12) \frac{d}{d x} [x^{2}]) - x^{2} (x^{2} - 12) \frac{d}{d x} [{(x + 2)}^{2} {(x - 2)}^{2}]}{{({(x + 2)}^{2} {(x - 2)}^{2})}^{2}}$

Step 2.1.2.3.4

Add $2 x$ and $0$ .

$\frac{{(x + 2)}^{2} {(x - 2)}^{2} (x^{2} (2 x) + (x^{2} - 12) \frac{d}{d x} [x^{2}]) - x^{2} (x^{2} - 12) \frac{d}{d x} [{(x + 2)}^{2} {(x - 2)}^{2}]}{{({(x + 2)}^{2} {(x - 2)}^{2})}^{2}}$

Step 2.1.2.4

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 2.1.2.4.1

Move $x$ .

$\frac{{(x + 2)}^{2} {(x - 2)}^{2} (x \cdot x^{2} \cdot 2 + (x^{2} - 12) \frac{d}{d x} [x^{2}]) - x^{2} (x^{2} - 12) \frac{d}{d x} [{(x + 2)}^{2} {(x - 2)}^{2}]}{{({(x + 2)}^{2} {(x - 2)}^{2})}^{2}}$

Step 2.1.2.4.2

Multiply $x$ by $x^{2}$ .

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Step 2.1.2.4.2.1

Raise $x$ to the power of $1$ .

$\frac{{(x + 2)}^{2} {(x - 2)}^{2} (x^{1} x^{2} \cdot 2 + (x^{2} - 12) \frac{d}{d x} [x^{2}]) - x^{2} (x^{2} - 12) \frac{d}{d x} [{(x + 2)}^{2} {(x - 2)}^{2}]}{{({(x + 2)}^{2} {(x - 2)}^{2})}^{2}}$

Step 2.1.2.4.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{{(x + 2)}^{2} {(x - 2)}^{2} (x^{1 + 2} \cdot 2 + (x^{2} - 12) \frac{d}{d x} [x^{2}]) - x^{2} (x^{2} - 12) \frac{d}{d x} [{(x + 2)}^{2} {(x - 2)}^{2}]}{{({(x + 2)}^{2} {(x - 2)}^{2})}^{2}}$

Step 2.1.2.4.3

Add $1$ and $2$ .

$\frac{{(x + 2)}^{2} {(x - 2)}^{2} (x^{3} \cdot 2 + (x^{2} - 12) \frac{d}{d x} [x^{2}]) - x^{2} (x^{2} - 12) \frac{d}{d x} [{(x + 2)}^{2} {(x - 2)}^{2}]}{{({(x + 2)}^{2} {(x - 2)}^{2})}^{2}}$

Step 2.1.2.5

Differentiate using the Power Rule.

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Step 2.1.2.5.1

Move $2$ to the left of $x^{3}$ .

$\frac{{(x + 2)}^{2} {(x - 2)}^{2} (2 \cdot x^{3} + (x^{2} - 12) \frac{d}{d x} [x^{2}]) - x^{2} (x^{2} - 12) \frac{d}{d x} [{(x + 2)}^{2} {(x - 2)}^{2}]}{{({(x + 2)}^{2} {(x - 2)}^{2})}^{2}}$

Step 2.1.2.5.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$\frac{{(x + 2)}^{2} {(x - 2)}^{2} (2 x^{3} + (x^{2} - 12) (2 x)) - x^{2} (x^{2} - 12) \frac{d}{d x} [{(x + 2)}^{2} {(x - 2)}^{2}]}{{({(x + 2)}^{2} {(x - 2)}^{2})}^{2}}$

Step 2.1.2.5.3

Move $2$ to the left of $x^{2} - 12$ .

$\frac{{(x + 2)}^{2} {(x - 2)}^{2} (2 x^{3} + 2 \cdot (x^{2} - 12) x) - x^{2} (x^{2} - 12) \frac{d}{d x} [{(x + 2)}^{2} {(x - 2)}^{2}]}{{({(x + 2)}^{2} {(x - 2)}^{2})}^{2}}$

Step 2.1.2.6

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = {(x + 2)}^{2}$ and $g (x) = {(x - 2)}^{2}$ .

$\frac{{(x + 2)}^{2} {(x - 2)}^{2} (2 x^{3} + 2 (x^{2} - 12) x) - x^{2} (x^{2} - 12) ({(x + 2)}^{2} \frac{d}{d x} [{(x - 2)}^{2}] + {(x - 2)}^{2} \frac{d}{d x} [{(x + 2)}^{2}])}{{({(x + 2)}^{2} {(x - 2)}^{2})}^{2}}$

Step 2.1.2.7

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = x - 2$ .

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Step 2.1.2.7.1

To apply the Chain Rule, set $u_{1}$ as $x - 2$ .

$\frac{{(x + 2)}^{2} {(x - 2)}^{2} (2 x^{3} + 2 (x^{2} - 12) x) - x^{2} (x^{2} - 12) ({(x + 2)}^{2} (\frac{d}{d u_{1}} [{u_{1}}^{2}] \frac{d}{d x} [x - 2]) + {(x - 2)}^{2} \frac{d}{d x} [{(x + 2)}^{2}])}{{({(x + 2)}^{2} {(x - 2)}^{2})}^{2}}$

Step 2.1.2.7.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 2$ .

$\frac{{(x + 2)}^{2} {(x - 2)}^{2} (2 x^{3} + 2 (x^{2} - 12) x) - x^{2} (x^{2} - 12) ({(x + 2)}^{2} (2 u_{1} \frac{d}{d x} [x - 2]) + {(x - 2)}^{2} \frac{d}{d x} [{(x + 2)}^{2}])}{{({(x + 2)}^{2} {(x - 2)}^{2})}^{2}}$

Step 2.1.2.7.3

Replace all occurrences of $u_{1}$ with $x - 2$ .

$\frac{{(x + 2)}^{2} {(x - 2)}^{2} (2 x^{3} + 2 (x^{2} - 12) x) - x^{2} (x^{2} - 12) ({(x + 2)}^{2} (2 (x - 2) \frac{d}{d x} [x - 2]) + {(x - 2)}^{2} \frac{d}{d x} [{(x + 2)}^{2}])}{{({(x + 2)}^{2} {(x - 2)}^{2})}^{2}}$

Step 2.1.2.8

Differentiate.

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Step 2.1.2.8.1

Move $2$ to the left of ${(x + 2)}^{2}$ .

$\frac{{(x + 2)}^{2} {(x - 2)}^{2} (2 x^{3} + 2 (x^{2} - 12) x) - x^{2} (x^{2} - 12) (2 \cdot {(x + 2)}^{2} (x - 2) \frac{d}{d x} [x - 2] + {(x - 2)}^{2} \frac{d}{d x} [{(x + 2)}^{2}])}{{({(x + 2)}^{2} {(x - 2)}^{2})}^{2}}$

Step 2.1.2.8.2

By the Sum Rule, the derivative of $x - 2$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 2]$ .

$\frac{{(x + 2)}^{2} {(x - 2)}^{2} (2 x^{3} + 2 (x^{2} - 12) x) - x^{2} (x^{2} - 12) (2 {(x + 2)}^{2} (x - 2) (\frac{d}{d x} [x] + \frac{d}{d x} [- 2]) + {(x - 2)}^{2} \frac{d}{d x} [{(x + 2)}^{2}])}{{({(x + 2)}^{2} {(x - 2)}^{2})}^{2}}$

Step 2.1.2.8.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$\frac{{(x + 2)}^{2} {(x - 2)}^{2} (2 x^{3} + 2 (x^{2} - 12) x) - x^{2} (x^{2} - 12) (2 {(x + 2)}^{2} (x - 2) (1 + \frac{d}{d x} [- 2]) + {(x - 2)}^{2} \frac{d}{d x} [{(x + 2)}^{2}])}{{({(x + 2)}^{2} {(x - 2)}^{2})}^{2}}$

Step 2.1.2.8.4

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$\frac{{(x + 2)}^{2} {(x - 2)}^{2} (2 x^{3} + 2 (x^{2} - 12) x) - x^{2} (x^{2} - 12) (2 {(x + 2)}^{2} (x - 2) (1 + 0) + {(x - 2)}^{2} \frac{d}{d x} [{(x + 2)}^{2}])}{{({(x + 2)}^{2} {(x - 2)}^{2})}^{2}}$

Step 2.1.2.8.5

Simplify the expression.

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Step 2.1.2.8.5.1

Add $1$ and $0$ .

$\frac{{(x + 2)}^{2} {(x - 2)}^{2} (2 x^{3} + 2 (x^{2} - 12) x) - x^{2} (x^{2} - 12) (2 {(x + 2)}^{2} (x - 2) \cdot 1 + {(x - 2)}^{2} \frac{d}{d x} [{(x + 2)}^{2}])}{{({(x + 2)}^{2} {(x - 2)}^{2})}^{2}}$

Step 2.1.2.8.5.2

Multiply $2$ by $1$ .

$\frac{{(x + 2)}^{2} {(x - 2)}^{2} (2 x^{3} + 2 (x^{2} - 12) x) - x^{2} (x^{2} - 12) (2 {(x + 2)}^{2} (x - 2) + {(x - 2)}^{2} \frac{d}{d x} [{(x + 2)}^{2}])}{{({(x + 2)}^{2} {(x - 2)}^{2})}^{2}}$

Step 2.1.2.9

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = x + 2$ .

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Step 2.1.2.9.1

To apply the Chain Rule, set $u_{2}$ as $x + 2$ .

$\frac{{(x + 2)}^{2} {(x - 2)}^{2} (2 x^{3} + 2 (x^{2} - 12) x) - x^{2} (x^{2} - 12) (2 {(x + 2)}^{2} (x - 2) + {(x - 2)}^{2} (\frac{d}{d u_{2}} [{u_{2}}^{2}] \frac{d}{d x} [x + 2]))}{{({(x + 2)}^{2} {(x - 2)}^{2})}^{2}}$

Step 2.1.2.9.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 2$ .

$\frac{{(x + 2)}^{2} {(x - 2)}^{2} (2 x^{3} + 2 (x^{2} - 12) x) - x^{2} (x^{2} - 12) (2 {(x + 2)}^{2} (x - 2) + {(x - 2)}^{2} (2 u_{2} \frac{d}{d x} [x + 2]))}{{({(x + 2)}^{2} {(x - 2)}^{2})}^{2}}$

Step 2.1.2.9.3

Replace all occurrences of $u_{2}$ with $x + 2$ .

$\frac{{(x + 2)}^{2} {(x - 2)}^{2} (2 x^{3} + 2 (x^{2} - 12) x) - x^{2} (x^{2} - 12) (2 {(x + 2)}^{2} (x - 2) + {(x - 2)}^{2} (2 (x + 2) \frac{d}{d x} [x + 2]))}{{({(x + 2)}^{2} {(x - 2)}^{2})}^{2}}$

Step 2.1.2.10

Differentiate.

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Step 2.1.2.10.1

Move $2$ to the left of ${(x - 2)}^{2}$ .

$\frac{{(x + 2)}^{2} {(x - 2)}^{2} (2 x^{3} + 2 (x^{2} - 12) x) - x^{2} (x^{2} - 12) (2 {(x + 2)}^{2} (x - 2) + 2 \cdot {(x - 2)}^{2} (x + 2) \frac{d}{d x} [x + 2])}{{({(x + 2)}^{2} {(x - 2)}^{2})}^{2}}$

Step 2.1.2.10.2

By the Sum Rule, the derivative of $x + 2$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [2]$ .

$\frac{{(x + 2)}^{2} {(x - 2)}^{2} (2 x^{3} + 2 (x^{2} - 12) x) - x^{2} (x^{2} - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2) (\frac{d}{d x} [x] + \frac{d}{d x} [2]))}{{({(x + 2)}^{2} {(x - 2)}^{2})}^{2}}$

Step 2.1.2.10.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$\frac{{(x + 2)}^{2} {(x - 2)}^{2} (2 x^{3} + 2 (x^{2} - 12) x) - x^{2} (x^{2} - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2) (1 + \frac{d}{d x} [2]))}{{({(x + 2)}^{2} {(x - 2)}^{2})}^{2}}$

Step 2.1.2.10.4

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$\frac{{(x + 2)}^{2} {(x - 2)}^{2} (2 x^{3} + 2 (x^{2} - 12) x) - x^{2} (x^{2} - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2) (1 + 0))}{{({(x + 2)}^{2} {(x - 2)}^{2})}^{2}}$

Step 2.1.2.10.5

Simplify the expression.

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Step 2.1.2.10.5.1

Add $1$ and $0$ .

$\frac{{(x + 2)}^{2} {(x - 2)}^{2} (2 x^{3} + 2 (x^{2} - 12) x) - x^{2} (x^{2} - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2) \cdot 1)}{{({(x + 2)}^{2} {(x - 2)}^{2})}^{2}}$

Step 2.1.2.10.5.2

Multiply $2$ by $1$ .

$\frac{{(x + 2)}^{2} {(x - 2)}^{2} (2 x^{3} + 2 (x^{2} - 12) x) - x^{2} (x^{2} - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2} {(x - 2)}^{2})}^{2}}$

Step 2.1.2.11

Simplify.

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Step 2.1.2.11.1

Apply the product rule to ${(x + 2)}^{2} {(x - 2)}^{2}$ .

$\frac{{(x + 2)}^{2} {(x - 2)}^{2} (2 x^{3} + 2 (x^{2} - 12) x) - x^{2} (x^{2} - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.2

Apply the distributive property.

$\frac{{(x + 2)}^{2} {(x - 2)}^{2} (2 x^{3} + (2 x^{2} + 2 \cdot - 12) x) - x^{2} (x^{2} - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.3

Apply the distributive property.

$\frac{{(x + 2)}^{2} {(x - 2)}^{2} (2 x^{3} + 2 x^{2} x + 2 \cdot - 12 x) - x^{2} (x^{2} - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.4

Apply the distributive property.

$\frac{{(x + 2)}^{2} {(x - 2)}^{2} (2 x^{3} + 2 x^{2} x + 2 \cdot - 12 x) + (- x^{2} x^{2} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5

Simplify the numerator.

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Step 2.1.2.11.5.1

Rewrite ${(x + 2)}^{2}$ as $(x + 2) (x + 2)$ .

$\frac{(x + 2) (x + 2) {(x - 2)}^{2} (2 x^{3} + 2 x^{2} x + 2 \cdot - 12 x) + (- x^{2} x^{2} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.2

Expand $(x + 2) (x + 2)$ using the FOIL Method.

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Step 2.1.2.11.5.2.1

Apply the distributive property.

$\frac{(x (x + 2) + 2 (x + 2)) {(x - 2)}^{2} (2 x^{3} + 2 x^{2} x + 2 \cdot - 12 x) + (- x^{2} x^{2} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.2.2

Apply the distributive property.

$\frac{(x \cdot x + x \cdot 2 + 2 (x + 2)) {(x - 2)}^{2} (2 x^{3} + 2 x^{2} x + 2 \cdot - 12 x) + (- x^{2} x^{2} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.2.3

Apply the distributive property.

$\frac{(x \cdot x + x \cdot 2 + 2 x + 2 \cdot 2) {(x - 2)}^{2} (2 x^{3} + 2 x^{2} x + 2 \cdot - 12 x) + (- x^{2} x^{2} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.3

Simplify and combine like terms.

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Step 2.1.2.11.5.3.1

Simplify each term.

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Step 2.1.2.11.5.3.1.1

Multiply $x$ by $x$ .

$\frac{(x^{2} + x \cdot 2 + 2 x + 2 \cdot 2) {(x - 2)}^{2} (2 x^{3} + 2 x^{2} x + 2 \cdot - 12 x) + (- x^{2} x^{2} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.3.1.2

Move $2$ to the left of $x$ .

$\frac{(x^{2} + 2 \cdot x + 2 x + 2 \cdot 2) {(x - 2)}^{2} (2 x^{3} + 2 x^{2} x + 2 \cdot - 12 x) + (- x^{2} x^{2} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.3.1.3

Multiply $2$ by $2$ .

$\frac{(x^{2} + 2 x + 2 x + 4) {(x - 2)}^{2} (2 x^{3} + 2 x^{2} x + 2 \cdot - 12 x) + (- x^{2} x^{2} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.3.2

Add $2 x$ and $2 x$ .

$\frac{(x^{2} + 4 x + 4) {(x - 2)}^{2} (2 x^{3} + 2 x^{2} x + 2 \cdot - 12 x) + (- x^{2} x^{2} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.4

Rewrite ${(x - 2)}^{2}$ as $(x - 2) (x - 2)$ .

$\frac{(x^{2} + 4 x + 4) ((x - 2) (x - 2)) (2 x^{3} + 2 x^{2} x + 2 \cdot - 12 x) + (- x^{2} x^{2} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.5

Expand $(x - 2) (x - 2)$ using the FOIL Method.

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Step 2.1.2.11.5.5.1

Apply the distributive property.

$\frac{(x^{2} + 4 x + 4) (x (x - 2) - 2 (x - 2)) (2 x^{3} + 2 x^{2} x + 2 \cdot - 12 x) + (- x^{2} x^{2} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.5.2

Apply the distributive property.

$\frac{(x^{2} + 4 x + 4) (x \cdot x + x \cdot - 2 - 2 (x - 2)) (2 x^{3} + 2 x^{2} x + 2 \cdot - 12 x) + (- x^{2} x^{2} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.5.3

Apply the distributive property.

$\frac{(x^{2} + 4 x + 4) (x \cdot x + x \cdot - 2 - 2 x - 2 \cdot - 2) (2 x^{3} + 2 x^{2} x + 2 \cdot - 12 x) + (- x^{2} x^{2} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.6

Simplify and combine like terms.

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Step 2.1.2.11.5.6.1

Simplify each term.

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Step 2.1.2.11.5.6.1.1

Multiply $x$ by $x$ .

$\frac{(x^{2} + 4 x + 4) (x^{2} + x \cdot - 2 - 2 x - 2 \cdot - 2) (2 x^{3} + 2 x^{2} x + 2 \cdot - 12 x) + (- x^{2} x^{2} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.6.1.2

Move $- 2$ to the left of $x$ .

$\frac{(x^{2} + 4 x + 4) (x^{2} - 2 \cdot x - 2 x - 2 \cdot - 2) (2 x^{3} + 2 x^{2} x + 2 \cdot - 12 x) + (- x^{2} x^{2} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.6.1.3

Multiply $- 2$ by $- 2$ .

$\frac{(x^{2} + 4 x + 4) (x^{2} - 2 x - 2 x + 4) (2 x^{3} + 2 x^{2} x + 2 \cdot - 12 x) + (- x^{2} x^{2} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.6.2

Subtract $2 x$ from $- 2 x$ .

$\frac{(x^{2} + 4 x + 4) (x^{2} - 4 x + 4) (2 x^{3} + 2 x^{2} x + 2 \cdot - 12 x) + (- x^{2} x^{2} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.7

Expand $(x^{2} + 4 x + 4) (x^{2} - 4 x + 4)$ by multiplying each term in the first expression by each term in the second expression.

$\frac{(x^{2} x^{2} + x^{2} (- 4 x) + x^{2} \cdot 4 + 4 x \cdot x^{2} + 4 x (- 4 x) + 4 x \cdot 4 + 4 x^{2} + 4 (- 4 x) + 4 \cdot 4) (2 x^{3} + 2 x^{2} x + 2 \cdot - 12 x) + (- x^{2} x^{2} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.8

Combine the opposite terms in $x^{2} x^{2} + x^{2} (- 4 x) + x^{2} \cdot 4 + 4 x \cdot x^{2} + 4 x (- 4 x) + 4 x \cdot 4 + 4 x^{2} + 4 (- 4 x) + 4 \cdot 4$ .

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Step 2.1.2.11.5.8.1

Reorder the factors in the terms $x^{2} (- 4 x)$ and $4 x \cdot x^{2}$ .

$\frac{(x^{2} x^{2} - 4 x^{2} x + x^{2} \cdot 4 + 4 x^{2} x + 4 x (- 4 x) + 4 x \cdot 4 + 4 x^{2} + 4 (- 4 x) + 4 \cdot 4) (2 x^{3} + 2 x^{2} x + 2 \cdot - 12 x) + (- x^{2} x^{2} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.8.2

Add $- 4 x^{2} x$ and $4 x^{2} x$ .

$\frac{(x^{2} x^{2} + x^{2} \cdot 4 + 0 + 4 x (- 4 x) + 4 x \cdot 4 + 4 x^{2} + 4 (- 4 x) + 4 \cdot 4) (2 x^{3} + 2 x^{2} x + 2 \cdot - 12 x) + (- x^{2} x^{2} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.8.3

Add $x^{2} x^{2} + x^{2} \cdot 4$ and $0$ .

$\frac{(x^{2} x^{2} + x^{2} \cdot 4 + 4 x (- 4 x) + 4 x \cdot 4 + 4 x^{2} + 4 (- 4 x) + 4 \cdot 4) (2 x^{3} + 2 x^{2} x + 2 \cdot - 12 x) + (- x^{2} x^{2} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.8.4

Reorder the factors in the terms $4 x \cdot 4$ and $4 (- 4 x)$ .

$\frac{(x^{2} x^{2} + x^{2} \cdot 4 + 4 x (- 4 x) + 4 \cdot 4 x + 4 x^{2} - 4 \cdot 4 x + 4 \cdot 4) (2 x^{3} + 2 x^{2} x + 2 \cdot - 12 x) + (- x^{2} x^{2} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.8.5

Subtract $4 \cdot 4 x$ from $4 \cdot 4 x$ .

$\frac{(x^{2} x^{2} + x^{2} \cdot 4 + 4 x (- 4 x) + 4 x^{2} + 0 + 4 \cdot 4) (2 x^{3} + 2 x^{2} x + 2 \cdot - 12 x) + (- x^{2} x^{2} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.8.6

Add $x^{2} x^{2} + x^{2} \cdot 4 + 4 x (- 4 x) + 4 x^{2}$ and $0$ .

$\frac{(x^{2} x^{2} + x^{2} \cdot 4 + 4 x (- 4 x) + 4 x^{2} + 4 \cdot 4) (2 x^{3} + 2 x^{2} x + 2 \cdot - 12 x) + (- x^{2} x^{2} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.9

Simplify each term.

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Step 2.1.2.11.5.9.1

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 2.1.2.11.5.9.1.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{(x^{2 + 2} + x^{2} \cdot 4 + 4 x (- 4 x) + 4 x^{2} + 4 \cdot 4) (2 x^{3} + 2 x^{2} x + 2 \cdot - 12 x) + (- x^{2} x^{2} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.9.1.2

Add $2$ and $2$ .

$\frac{(x^{4} + x^{2} \cdot 4 + 4 x (- 4 x) + 4 x^{2} + 4 \cdot 4) (2 x^{3} + 2 x^{2} x + 2 \cdot - 12 x) + (- x^{2} x^{2} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.9.2

Move $4$ to the left of $x^{2}$ .

$\frac{(x^{4} + 4 \cdot x^{2} + 4 x (- 4 x) + 4 x^{2} + 4 \cdot 4) (2 x^{3} + 2 x^{2} x + 2 \cdot - 12 x) + (- x^{2} x^{2} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.9.3

Rewrite using the commutative property of multiplication.

$\frac{(x^{4} + 4 x^{2} + 4 \cdot - 4 x \cdot x + 4 x^{2} + 4 \cdot 4) (2 x^{3} + 2 x^{2} x + 2 \cdot - 12 x) + (- x^{2} x^{2} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.9.4

Multiply $x$ by $x$ by adding the exponents.

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Step 2.1.2.11.5.9.4.1

Move $x$ .

$\frac{(x^{4} + 4 x^{2} + 4 \cdot - 4 (x \cdot x) + 4 x^{2} + 4 \cdot 4) (2 x^{3} + 2 x^{2} x + 2 \cdot - 12 x) + (- x^{2} x^{2} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.9.4.2

Multiply $x$ by $x$ .

$\frac{(x^{4} + 4 x^{2} + 4 \cdot - 4 x^{2} + 4 x^{2} + 4 \cdot 4) (2 x^{3} + 2 x^{2} x + 2 \cdot - 12 x) + (- x^{2} x^{2} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.9.5

Multiply $4$ by $- 4$ .

$\frac{(x^{4} + 4 x^{2} - 16 x^{2} + 4 x^{2} + 4 \cdot 4) (2 x^{3} + 2 x^{2} x + 2 \cdot - 12 x) + (- x^{2} x^{2} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.9.6

Multiply $4$ by $4$ .

$\frac{(x^{4} + 4 x^{2} - 16 x^{2} + 4 x^{2} + 16) (2 x^{3} + 2 x^{2} x + 2 \cdot - 12 x) + (- x^{2} x^{2} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.10

Subtract $16 x^{2}$ from $4 x^{2}$ .

$\frac{(x^{4} - 12 x^{2} + 4 x^{2} + 16) (2 x^{3} + 2 x^{2} x + 2 \cdot - 12 x) + (- x^{2} x^{2} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.11

Add $- 12 x^{2}$ and $4 x^{2}$ .

$\frac{(x^{4} - 8 x^{2} + 16) (2 x^{3} + 2 x^{2} x + 2 \cdot - 12 x) + (- x^{2} x^{2} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.12

Simplify each term.

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Step 2.1.2.11.5.12.1

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 2.1.2.11.5.12.1.1

Move $x$ .

$\frac{(x^{4} - 8 x^{2} + 16) (2 x^{3} + 2 (x \cdot x^{2}) + 2 \cdot - 12 x) + (- x^{2} x^{2} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.12.1.2

Multiply $x$ by $x^{2}$ .

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Step 2.1.2.11.5.12.1.2.1

Raise $x$ to the power of $1$ .

$\frac{(x^{4} - 8 x^{2} + 16) (2 x^{3} + 2 (x^{1} x^{2}) + 2 \cdot - 12 x) + (- x^{2} x^{2} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.12.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{(x^{4} - 8 x^{2} + 16) (2 x^{3} + 2 x^{1 + 2} + 2 \cdot - 12 x) + (- x^{2} x^{2} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.12.1.3

Add $1$ and $2$ .

$\frac{(x^{4} - 8 x^{2} + 16) (2 x^{3} + 2 x^{3} + 2 \cdot - 12 x) + (- x^{2} x^{2} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.12.2

Multiply $2$ by $- 12$ .

$\frac{(x^{4} - 8 x^{2} + 16) (2 x^{3} + 2 x^{3} - 24 x) + (- x^{2} x^{2} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.13

Add $2 x^{3}$ and $2 x^{3}$ .

$\frac{(x^{4} - 8 x^{2} + 16) (4 x^{3} - 24 x) + (- x^{2} x^{2} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.14

Expand $(x^{4} - 8 x^{2} + 16) (4 x^{3} - 24 x)$ by multiplying each term in the first expression by each term in the second expression.

$\frac{x^{4} (4 x^{3}) + x^{4} (- 24 x) - 8 x^{2} (4 x^{3}) - 8 x^{2} (- 24 x) + 16 (4 x^{3}) + 16 (- 24 x) + (- x^{2} x^{2} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.15

Simplify each term.

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Step 2.1.2.11.5.15.1

Rewrite using the commutative property of multiplication.

$\frac{4 x^{4} x^{3} + x^{4} (- 24 x) - 8 x^{2} (4 x^{3}) - 8 x^{2} (- 24 x) + 16 (4 x^{3}) + 16 (- 24 x) + (- x^{2} x^{2} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.15.2

Multiply $x^{4}$ by $x^{3}$ by adding the exponents.

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Step 2.1.2.11.5.15.2.1

Move $x^{3}$ .

$\frac{4 (x^{3} x^{4}) + x^{4} (- 24 x) - 8 x^{2} (4 x^{3}) - 8 x^{2} (- 24 x) + 16 (4 x^{3}) + 16 (- 24 x) + (- x^{2} x^{2} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.15.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{4 x^{3 + 4} + x^{4} (- 24 x) - 8 x^{2} (4 x^{3}) - 8 x^{2} (- 24 x) + 16 (4 x^{3}) + 16 (- 24 x) + (- x^{2} x^{2} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.15.2.3

Add $3$ and $4$ .

$\frac{4 x^{7} + x^{4} (- 24 x) - 8 x^{2} (4 x^{3}) - 8 x^{2} (- 24 x) + 16 (4 x^{3}) + 16 (- 24 x) + (- x^{2} x^{2} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.15.3

Rewrite using the commutative property of multiplication.

$\frac{4 x^{7} - 24 x^{4} x - 8 x^{2} (4 x^{3}) - 8 x^{2} (- 24 x) + 16 (4 x^{3}) + 16 (- 24 x) + (- x^{2} x^{2} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.15.4

Multiply $x^{4}$ by $x$ by adding the exponents.

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Step 2.1.2.11.5.15.4.1

Move $x$ .

$\frac{4 x^{7} - 24 (x \cdot x^{4}) - 8 x^{2} (4 x^{3}) - 8 x^{2} (- 24 x) + 16 (4 x^{3}) + 16 (- 24 x) + (- x^{2} x^{2} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.15.4.2

Multiply $x$ by $x^{4}$ .

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Step 2.1.2.11.5.15.4.2.1

Raise $x$ to the power of $1$ .

$\frac{4 x^{7} - 24 (x^{1} x^{4}) - 8 x^{2} (4 x^{3}) - 8 x^{2} (- 24 x) + 16 (4 x^{3}) + 16 (- 24 x) + (- x^{2} x^{2} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.15.4.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{4 x^{7} - 24 x^{1 + 4} - 8 x^{2} (4 x^{3}) - 8 x^{2} (- 24 x) + 16 (4 x^{3}) + 16 (- 24 x) + (- x^{2} x^{2} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.15.4.3

Add $1$ and $4$ .

$\frac{4 x^{7} - 24 x^{5} - 8 x^{2} (4 x^{3}) - 8 x^{2} (- 24 x) + 16 (4 x^{3}) + 16 (- 24 x) + (- x^{2} x^{2} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.15.5

Rewrite using the commutative property of multiplication.

$\frac{4 x^{7} - 24 x^{5} - 8 \cdot 4 x^{2} x^{3} - 8 x^{2} (- 24 x) + 16 (4 x^{3}) + 16 (- 24 x) + (- x^{2} x^{2} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.15.6

Multiply $x^{2}$ by $x^{3}$ by adding the exponents.

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Step 2.1.2.11.5.15.6.1

Move $x^{3}$ .

$\frac{4 x^{7} - 24 x^{5} - 8 \cdot 4 (x^{3} x^{2}) - 8 x^{2} (- 24 x) + 16 (4 x^{3}) + 16 (- 24 x) + (- x^{2} x^{2} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.15.6.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{4 x^{7} - 24 x^{5} - 8 \cdot 4 x^{3 + 2} - 8 x^{2} (- 24 x) + 16 (4 x^{3}) + 16 (- 24 x) + (- x^{2} x^{2} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.15.6.3

Add $3$ and $2$ .

$\frac{4 x^{7} - 24 x^{5} - 8 \cdot 4 x^{5} - 8 x^{2} (- 24 x) + 16 (4 x^{3}) + 16 (- 24 x) + (- x^{2} x^{2} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.15.7

Multiply $- 8$ by $4$ .

$\frac{4 x^{7} - 24 x^{5} - 32 x^{5} - 8 x^{2} (- 24 x) + 16 (4 x^{3}) + 16 (- 24 x) + (- x^{2} x^{2} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.15.8

Rewrite using the commutative property of multiplication.

$\frac{4 x^{7} - 24 x^{5} - 32 x^{5} - 8 \cdot - 24 x^{2} x + 16 (4 x^{3}) + 16 (- 24 x) + (- x^{2} x^{2} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.15.9

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 2.1.2.11.5.15.9.1

Move $x$ .

$\frac{4 x^{7} - 24 x^{5} - 32 x^{5} - 8 \cdot - 24 (x \cdot x^{2}) + 16 (4 x^{3}) + 16 (- 24 x) + (- x^{2} x^{2} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.15.9.2

Multiply $x$ by $x^{2}$ .

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Step 2.1.2.11.5.15.9.2.1

Raise $x$ to the power of $1$ .

$\frac{4 x^{7} - 24 x^{5} - 32 x^{5} - 8 \cdot - 24 (x^{1} x^{2}) + 16 (4 x^{3}) + 16 (- 24 x) + (- x^{2} x^{2} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.15.9.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{4 x^{7} - 24 x^{5} - 32 x^{5} - 8 \cdot - 24 x^{1 + 2} + 16 (4 x^{3}) + 16 (- 24 x) + (- x^{2} x^{2} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.15.9.3

Add $1$ and $2$ .

$\frac{4 x^{7} - 24 x^{5} - 32 x^{5} - 8 \cdot - 24 x^{3} + 16 (4 x^{3}) + 16 (- 24 x) + (- x^{2} x^{2} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.15.10

Multiply $- 8$ by $- 24$ .

$\frac{4 x^{7} - 24 x^{5} - 32 x^{5} + 192 x^{3} + 16 (4 x^{3}) + 16 (- 24 x) + (- x^{2} x^{2} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.15.11

Multiply $4$ by $16$ .

$\frac{4 x^{7} - 24 x^{5} - 32 x^{5} + 192 x^{3} + 64 x^{3} + 16 (- 24 x) + (- x^{2} x^{2} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.15.12

Multiply $- 24$ by $16$ .

$\frac{4 x^{7} - 24 x^{5} - 32 x^{5} + 192 x^{3} + 64 x^{3} - 384 x + (- x^{2} x^{2} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.16

Subtract $32 x^{5}$ from $- 24 x^{5}$ .

$\frac{4 x^{7} - 56 x^{5} + 192 x^{3} + 64 x^{3} - 384 x + (- x^{2} x^{2} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.17

Add $192 x^{3}$ and $64 x^{3}$ .

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x + (- x^{2} x^{2} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.18

Simplify each term.

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Step 2.1.2.11.5.18.1

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 2.1.2.11.5.18.1.1

Move $x^{2}$ .

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x + (- (x^{2} x^{2}) - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.18.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x + (- x^{2 + 2} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.18.1.3

Add $2$ and $2$ .

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x + (- x^{4} - x^{2} \cdot - 12) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.18.2

Multiply $- 12$ by $- 1$ .

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x + (- x^{4} + 12 x^{2}) (2 {(x + 2)}^{2} (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.19

Simplify each term.

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Step 2.1.2.11.5.19.1

Rewrite ${(x + 2)}^{2}$ as $(x + 2) (x + 2)$ .

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x + (- x^{4} + 12 x^{2}) (2 ((x + 2) (x + 2)) (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.19.2

Expand $(x + 2) (x + 2)$ using the FOIL Method.

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Step 2.1.2.11.5.19.2.1

Apply the distributive property.

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x + (- x^{4} + 12 x^{2}) (2 (x (x + 2) + 2 (x + 2)) (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.19.2.2

Apply the distributive property.

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x + (- x^{4} + 12 x^{2}) (2 (x \cdot x + x \cdot 2 + 2 (x + 2)) (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.19.2.3

Apply the distributive property.

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x + (- x^{4} + 12 x^{2}) (2 (x \cdot x + x \cdot 2 + 2 x + 2 \cdot 2) (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.19.3

Simplify and combine like terms.

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Step 2.1.2.11.5.19.3.1

Simplify each term.

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Step 2.1.2.11.5.19.3.1.1

Multiply $x$ by $x$ .

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x + (- x^{4} + 12 x^{2}) (2 (x^{2} + x \cdot 2 + 2 x + 2 \cdot 2) (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.19.3.1.2

Move $2$ to the left of $x$ .

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x + (- x^{4} + 12 x^{2}) (2 (x^{2} + 2 \cdot x + 2 x + 2 \cdot 2) (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.19.3.1.3

Multiply $2$ by $2$ .

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x + (- x^{4} + 12 x^{2}) (2 (x^{2} + 2 x + 2 x + 4) (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.19.3.2

Add $2 x$ and $2 x$ .

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x + (- x^{4} + 12 x^{2}) (2 (x^{2} + 4 x + 4) (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.19.4

Apply the distributive property.

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x + (- x^{4} + 12 x^{2}) ((2 x^{2} + 2 (4 x) + 2 \cdot 4) (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.19.5

Simplify.

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Step 2.1.2.11.5.19.5.1

Multiply $4$ by $2$ .

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x + (- x^{4} + 12 x^{2}) ((2 x^{2} + 8 x + 2 \cdot 4) (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.19.5.2

Multiply $2$ by $4$ .

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x + (- x^{4} + 12 x^{2}) ((2 x^{2} + 8 x + 8) (x - 2) + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.19.6

Expand $(2 x^{2} + 8 x + 8) (x - 2)$ by multiplying each term in the first expression by each term in the second expression.

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x + (- x^{4} + 12 x^{2}) (2 x^{2} x + 2 x^{2} \cdot - 2 + 8 x \cdot x + 8 x \cdot - 2 + 8 x + 8 \cdot - 2 + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.19.7

Simplify each term.

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Step 2.1.2.11.5.19.7.1

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 2.1.2.11.5.19.7.1.1

Move $x$ .

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x + (- x^{4} + 12 x^{2}) (2 (x \cdot x^{2}) + 2 x^{2} \cdot - 2 + 8 x \cdot x + 8 x \cdot - 2 + 8 x + 8 \cdot - 2 + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.19.7.1.2

Multiply $x$ by $x^{2}$ .

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Step 2.1.2.11.5.19.7.1.2.1

Raise $x$ to the power of $1$ .

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x + (- x^{4} + 12 x^{2}) (2 (x^{1} x^{2}) + 2 x^{2} \cdot - 2 + 8 x \cdot x + 8 x \cdot - 2 + 8 x + 8 \cdot - 2 + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.19.7.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x + (- x^{4} + 12 x^{2}) (2 x^{1 + 2} + 2 x^{2} \cdot - 2 + 8 x \cdot x + 8 x \cdot - 2 + 8 x + 8 \cdot - 2 + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.19.7.1.3

Add $1$ and $2$ .

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x + (- x^{4} + 12 x^{2}) (2 x^{3} + 2 x^{2} \cdot - 2 + 8 x \cdot x + 8 x \cdot - 2 + 8 x + 8 \cdot - 2 + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.19.7.2

Multiply $- 2$ by $2$ .

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x + (- x^{4} + 12 x^{2}) (2 x^{3} - 4 x^{2} + 8 x \cdot x + 8 x \cdot - 2 + 8 x + 8 \cdot - 2 + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.19.7.3

Multiply $x$ by $x$ by adding the exponents.

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Step 2.1.2.11.5.19.7.3.1

Move $x$ .

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x + (- x^{4} + 12 x^{2}) (2 x^{3} - 4 x^{2} + 8 (x \cdot x) + 8 x \cdot - 2 + 8 x + 8 \cdot - 2 + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.19.7.3.2

Multiply $x$ by $x$ .

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x + (- x^{4} + 12 x^{2}) (2 x^{3} - 4 x^{2} + 8 x^{2} + 8 x \cdot - 2 + 8 x + 8 \cdot - 2 + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.19.7.4

Multiply $- 2$ by $8$ .

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x + (- x^{4} + 12 x^{2}) (2 x^{3} - 4 x^{2} + 8 x^{2} - 16 x + 8 x + 8 \cdot - 2 + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.19.7.5

Multiply $8$ by $- 2$ .

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x + (- x^{4} + 12 x^{2}) (2 x^{3} - 4 x^{2} + 8 x^{2} - 16 x + 8 x - 16 + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.19.8

Add $- 4 x^{2}$ and $8 x^{2}$ .

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x + (- x^{4} + 12 x^{2}) (2 x^{3} + 4 x^{2} - 16 x + 8 x - 16 + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.19.9

Add $- 16 x$ and $8 x$ .

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x + (- x^{4} + 12 x^{2}) (2 x^{3} + 4 x^{2} - 8 x - 16 + 2 {(x - 2)}^{2} (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.19.10

Rewrite ${(x - 2)}^{2}$ as $(x - 2) (x - 2)$ .

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x + (- x^{4} + 12 x^{2}) (2 x^{3} + 4 x^{2} - 8 x - 16 + 2 ((x - 2) (x - 2)) (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.19.11

Expand $(x - 2) (x - 2)$ using the FOIL Method.

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Step 2.1.2.11.5.19.11.1

Apply the distributive property.

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x + (- x^{4} + 12 x^{2}) (2 x^{3} + 4 x^{2} - 8 x - 16 + 2 (x (x - 2) - 2 (x - 2)) (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.19.11.2

Apply the distributive property.

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x + (- x^{4} + 12 x^{2}) (2 x^{3} + 4 x^{2} - 8 x - 16 + 2 (x \cdot x + x \cdot - 2 - 2 (x - 2)) (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.19.11.3

Apply the distributive property.

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x + (- x^{4} + 12 x^{2}) (2 x^{3} + 4 x^{2} - 8 x - 16 + 2 (x \cdot x + x \cdot - 2 - 2 x - 2 \cdot - 2) (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.19.12

Simplify and combine like terms.

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Step 2.1.2.11.5.19.12.1

Simplify each term.

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Step 2.1.2.11.5.19.12.1.1

Multiply $x$ by $x$ .

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x + (- x^{4} + 12 x^{2}) (2 x^{3} + 4 x^{2} - 8 x - 16 + 2 (x^{2} + x \cdot - 2 - 2 x - 2 \cdot - 2) (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.19.12.1.2

Move $- 2$ to the left of $x$ .

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x + (- x^{4} + 12 x^{2}) (2 x^{3} + 4 x^{2} - 8 x - 16 + 2 (x^{2} - 2 \cdot x - 2 x - 2 \cdot - 2) (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.19.12.1.3

Multiply $- 2$ by $- 2$ .

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x + (- x^{4} + 12 x^{2}) (2 x^{3} + 4 x^{2} - 8 x - 16 + 2 (x^{2} - 2 x - 2 x + 4) (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.19.12.2

Subtract $2 x$ from $- 2 x$ .

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x + (- x^{4} + 12 x^{2}) (2 x^{3} + 4 x^{2} - 8 x - 16 + 2 (x^{2} - 4 x + 4) (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.19.13

Apply the distributive property.

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x + (- x^{4} + 12 x^{2}) (2 x^{3} + 4 x^{2} - 8 x - 16 + (2 x^{2} + 2 (- 4 x) + 2 \cdot 4) (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.19.14

Simplify.

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Step 2.1.2.11.5.19.14.1

Multiply $- 4$ by $2$ .

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x + (- x^{4} + 12 x^{2}) (2 x^{3} + 4 x^{2} - 8 x - 16 + (2 x^{2} - 8 x + 2 \cdot 4) (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.19.14.2

Multiply $2$ by $4$ .

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x + (- x^{4} + 12 x^{2}) (2 x^{3} + 4 x^{2} - 8 x - 16 + (2 x^{2} - 8 x + 8) (x + 2))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.19.15

Expand $(2 x^{2} - 8 x + 8) (x + 2)$ by multiplying each term in the first expression by each term in the second expression.

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x + (- x^{4} + 12 x^{2}) (2 x^{3} + 4 x^{2} - 8 x - 16 + 2 x^{2} x + 2 x^{2} \cdot 2 - 8 x \cdot x - 8 x \cdot 2 + 8 x + 8 \cdot 2)}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.19.16

Simplify each term.

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Step 2.1.2.11.5.19.16.1

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 2.1.2.11.5.19.16.1.1

Move $x$ .

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x + (- x^{4} + 12 x^{2}) (2 x^{3} + 4 x^{2} - 8 x - 16 + 2 (x \cdot x^{2}) + 2 x^{2} \cdot 2 - 8 x \cdot x - 8 x \cdot 2 + 8 x + 8 \cdot 2)}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.19.16.1.2

Multiply $x$ by $x^{2}$ .

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Step 2.1.2.11.5.19.16.1.2.1

Raise $x$ to the power of $1$ .

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x + (- x^{4} + 12 x^{2}) (2 x^{3} + 4 x^{2} - 8 x - 16 + 2 (x^{1} x^{2}) + 2 x^{2} \cdot 2 - 8 x \cdot x - 8 x \cdot 2 + 8 x + 8 \cdot 2)}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.19.16.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x + (- x^{4} + 12 x^{2}) (2 x^{3} + 4 x^{2} - 8 x - 16 + 2 x^{1 + 2} + 2 x^{2} \cdot 2 - 8 x \cdot x - 8 x \cdot 2 + 8 x + 8 \cdot 2)}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.19.16.1.3

Add $1$ and $2$ .

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x + (- x^{4} + 12 x^{2}) (2 x^{3} + 4 x^{2} - 8 x - 16 + 2 x^{3} + 2 x^{2} \cdot 2 - 8 x \cdot x - 8 x \cdot 2 + 8 x + 8 \cdot 2)}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.19.16.2

Multiply $2$ by $2$ .

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x + (- x^{4} + 12 x^{2}) (2 x^{3} + 4 x^{2} - 8 x - 16 + 2 x^{3} + 4 x^{2} - 8 x \cdot x - 8 x \cdot 2 + 8 x + 8 \cdot 2)}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.19.16.3

Multiply $x$ by $x$ by adding the exponents.

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Step 2.1.2.11.5.19.16.3.1

Move $x$ .

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x + (- x^{4} + 12 x^{2}) (2 x^{3} + 4 x^{2} - 8 x - 16 + 2 x^{3} + 4 x^{2} - 8 (x \cdot x) - 8 x \cdot 2 + 8 x + 8 \cdot 2)}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.19.16.3.2

Multiply $x$ by $x$ .

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x + (- x^{4} + 12 x^{2}) (2 x^{3} + 4 x^{2} - 8 x - 16 + 2 x^{3} + 4 x^{2} - 8 x^{2} - 8 x \cdot 2 + 8 x + 8 \cdot 2)}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.19.16.4

Multiply $2$ by $- 8$ .

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x + (- x^{4} + 12 x^{2}) (2 x^{3} + 4 x^{2} - 8 x - 16 + 2 x^{3} + 4 x^{2} - 8 x^{2} - 16 x + 8 x + 8 \cdot 2)}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.19.16.5

Multiply $8$ by $2$ .

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x + (- x^{4} + 12 x^{2}) (2 x^{3} + 4 x^{2} - 8 x - 16 + 2 x^{3} + 4 x^{2} - 8 x^{2} - 16 x + 8 x + 16)}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.19.17

Subtract $8 x^{2}$ from $4 x^{2}$ .

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x + (- x^{4} + 12 x^{2}) (2 x^{3} + 4 x^{2} - 8 x - 16 + 2 x^{3} - 4 x^{2} - 16 x + 8 x + 16)}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.19.18

Add $- 16 x$ and $8 x$ .

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x + (- x^{4} + 12 x^{2}) (2 x^{3} + 4 x^{2} - 8 x - 16 + 2 x^{3} - 4 x^{2} - 8 x + 16)}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.20

Combine the opposite terms in $2 x^{3} + 4 x^{2} - 8 x - 16 + 2 x^{3} - 4 x^{2} - 8 x + 16$ .

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Step 2.1.2.11.5.20.1

Subtract $4 x^{2}$ from $4 x^{2}$ .

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x + (- x^{4} + 12 x^{2}) (2 x^{3} - 8 x - 16 + 2 x^{3} + 0 - 8 x + 16)}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.20.2

Add $2 x^{3} - 8 x - 16 + 2 x^{3}$ and $0$ .

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x + (- x^{4} + 12 x^{2}) (2 x^{3} - 8 x - 16 + 2 x^{3} - 8 x + 16)}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.20.3

Add $- 16$ and $16$ .

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x + (- x^{4} + 12 x^{2}) (2 x^{3} - 8 x + 2 x^{3} - 8 x + 0)}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.20.4

Add $2 x^{3} - 8 x + 2 x^{3} - 8 x$ and $0$ .

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x + (- x^{4} + 12 x^{2}) (2 x^{3} - 8 x + 2 x^{3} - 8 x)}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.21

Add $2 x^{3}$ and $2 x^{3}$ .

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x + (- x^{4} + 12 x^{2}) (4 x^{3} - 8 x - 8 x)}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.22

Subtract $8 x$ from $- 8 x$ .

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x + (- x^{4} + 12 x^{2}) (4 x^{3} - 16 x)}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.23

Expand $(- x^{4} + 12 x^{2}) (4 x^{3} - 16 x)$ using the FOIL Method.

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Step 2.1.2.11.5.23.1

Apply the distributive property.

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x - x^{4} (4 x^{3} - 16 x) + 12 x^{2} (4 x^{3} - 16 x)}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.23.2

Apply the distributive property.

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x - x^{4} (4 x^{3}) - x^{4} (- 16 x) + 12 x^{2} (4 x^{3} - 16 x)}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.23.3

Apply the distributive property.

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x - x^{4} (4 x^{3}) - x^{4} (- 16 x) + 12 x^{2} (4 x^{3}) + 12 x^{2} (- 16 x)}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.24

Simplify and combine like terms.

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Step 2.1.2.11.5.24.1

Simplify each term.

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Step 2.1.2.11.5.24.1.1

Rewrite using the commutative property of multiplication.

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x - 1 \cdot 4 x^{4} x^{3} - x^{4} (- 16 x) + 12 x^{2} (4 x^{3}) + 12 x^{2} (- 16 x)}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.24.1.2

Multiply $x^{4}$ by $x^{3}$ by adding the exponents.

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Step 2.1.2.11.5.24.1.2.1

Move $x^{3}$ .

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x - 1 \cdot 4 (x^{3} x^{4}) - x^{4} (- 16 x) + 12 x^{2} (4 x^{3}) + 12 x^{2} (- 16 x)}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.24.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x - 1 \cdot 4 x^{3 + 4} - x^{4} (- 16 x) + 12 x^{2} (4 x^{3}) + 12 x^{2} (- 16 x)}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.24.1.2.3

Add $3$ and $4$ .

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x - 1 \cdot 4 x^{7} - x^{4} (- 16 x) + 12 x^{2} (4 x^{3}) + 12 x^{2} (- 16 x)}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.24.1.3

Multiply $- 1$ by $4$ .

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x - 4 x^{7} - x^{4} (- 16 x) + 12 x^{2} (4 x^{3}) + 12 x^{2} (- 16 x)}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.24.1.4

Rewrite using the commutative property of multiplication.

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x - 4 x^{7} - 1 \cdot - 16 x^{4} x + 12 x^{2} (4 x^{3}) + 12 x^{2} (- 16 x)}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.24.1.5

Multiply $x^{4}$ by $x$ by adding the exponents.

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Step 2.1.2.11.5.24.1.5.1

Move $x$ .

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x - 4 x^{7} - 1 \cdot - 16 (x \cdot x^{4}) + 12 x^{2} (4 x^{3}) + 12 x^{2} (- 16 x)}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.24.1.5.2

Multiply $x$ by $x^{4}$ .

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Step 2.1.2.11.5.24.1.5.2.1

Raise $x$ to the power of $1$ .

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x - 4 x^{7} - 1 \cdot - 16 (x^{1} x^{4}) + 12 x^{2} (4 x^{3}) + 12 x^{2} (- 16 x)}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.24.1.5.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x - 4 x^{7} - 1 \cdot - 16 x^{1 + 4} + 12 x^{2} (4 x^{3}) + 12 x^{2} (- 16 x)}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.24.1.5.3

Add $1$ and $4$ .

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x - 4 x^{7} - 1 \cdot - 16 x^{5} + 12 x^{2} (4 x^{3}) + 12 x^{2} (- 16 x)}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.24.1.6

Multiply $- 1$ by $- 16$ .

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x - 4 x^{7} + 16 x^{5} + 12 x^{2} (4 x^{3}) + 12 x^{2} (- 16 x)}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.24.1.7

Rewrite using the commutative property of multiplication.

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x - 4 x^{7} + 16 x^{5} + 12 \cdot 4 x^{2} x^{3} + 12 x^{2} (- 16 x)}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.24.1.8

Multiply $x^{2}$ by $x^{3}$ by adding the exponents.

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Step 2.1.2.11.5.24.1.8.1

Move $x^{3}$ .

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x - 4 x^{7} + 16 x^{5} + 12 \cdot 4 (x^{3} x^{2}) + 12 x^{2} (- 16 x)}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.24.1.8.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x - 4 x^{7} + 16 x^{5} + 12 \cdot 4 x^{3 + 2} + 12 x^{2} (- 16 x)}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.24.1.8.3

Add $3$ and $2$ .

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x - 4 x^{7} + 16 x^{5} + 12 \cdot 4 x^{5} + 12 x^{2} (- 16 x)}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.24.1.9

Multiply $12$ by $4$ .

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x - 4 x^{7} + 16 x^{5} + 48 x^{5} + 12 x^{2} (- 16 x)}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.24.1.10

Rewrite using the commutative property of multiplication.

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x - 4 x^{7} + 16 x^{5} + 48 x^{5} + 12 \cdot - 16 x^{2} x}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.24.1.11

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 2.1.2.11.5.24.1.11.1

Move $x$ .

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x - 4 x^{7} + 16 x^{5} + 48 x^{5} + 12 \cdot - 16 (x \cdot x^{2})}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.24.1.11.2

Multiply $x$ by $x^{2}$ .

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Step 2.1.2.11.5.24.1.11.2.1

Raise $x$ to the power of $1$ .

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x - 4 x^{7} + 16 x^{5} + 48 x^{5} + 12 \cdot - 16 (x^{1} x^{2})}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.24.1.11.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x - 4 x^{7} + 16 x^{5} + 48 x^{5} + 12 \cdot - 16 x^{1 + 2}}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.24.1.11.3

Add $1$ and $2$ .

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x - 4 x^{7} + 16 x^{5} + 48 x^{5} + 12 \cdot - 16 x^{3}}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.24.1.12

Multiply $12$ by $- 16$ .

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x - 4 x^{7} + 16 x^{5} + 48 x^{5} - 192 x^{3}}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.24.2

Add $16 x^{5}$ and $48 x^{5}$ .

$\frac{4 x^{7} - 56 x^{5} + 256 x^{3} - 384 x - 4 x^{7} + 64 x^{5} - 192 x^{3}}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.25

Subtract $4 x^{7}$ from $4 x^{7}$ .

$\frac{- 56 x^{5} + 256 x^{3} - 384 x + 0 + 64 x^{5} - 192 x^{3}}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.26

Add $- 56 x^{5}$ and $0$ .

$\frac{- 56 x^{5} + 256 x^{3} - 384 x + 64 x^{5} - 192 x^{3}}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.27

Add $- 56 x^{5}$ and $64 x^{5}$ .

$\frac{8 x^{5} + 256 x^{3} - 384 x - 192 x^{3}}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.28

Subtract $192 x^{3}$ from $256 x^{3}$ .

$\frac{8 x^{5} + 64 x^{3} - 384 x}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.29

Rewrite $8 x^{5} + 64 x^{3} - 384 x$ in a factored form.

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Step 2.1.2.11.5.29.1

Factor $8 x$ out of $8 x^{5} + 64 x^{3} - 384 x$ .

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Step 2.1.2.11.5.29.1.1

Factor $8 x$ out of $8 x^{5}$ .

$\frac{8 x (x^{4}) + 64 x^{3} - 384 x}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.29.1.2

Factor $8 x$ out of $64 x^{3}$ .

$\frac{8 x (x^{4}) + 8 x (8 x^{2}) - 384 x}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.29.1.3

Factor $8 x$ out of $- 384 x$ .

$\frac{8 x (x^{4}) + 8 x (8 x^{2}) + 8 x (- 48)}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.29.1.4

Factor $8 x$ out of $8 x (x^{4}) + 8 x (8 x^{2})$ .

$\frac{8 x (x^{4} + 8 x^{2}) + 8 x (- 48)}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.29.1.5

Factor $8 x$ out of $8 x (x^{4} + 8 x^{2}) + 8 x (- 48)$ .

$\frac{8 x (x^{4} + 8 x^{2} - 48)}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.29.2

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$\frac{8 x ({(x^{2})}^{2} + 8 x^{2} - 48)}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.29.3

Let $u_{3} = x^{2}$ . Substitute $u_{3}$ for all occurrences of $x^{2}$ .

$\frac{8 x ({u_{3}}^{2} + 8 u_{3} - 48)}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.29.4

Factor ${u_{3}}^{2} + 8 u_{3} - 48$ using the AC method.

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Step 2.1.2.11.5.29.4.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 48$ and whose sum is $8$ .

$- 4, 12$

Step 2.1.2.11.5.29.4.2

Write the factored form using these integers.

$\frac{8 x ((u_{3} - 4) (u_{3} + 12))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.29.5

Replace all occurrences of $u_{3}$ with $x^{2}$ .

$\frac{8 x ((x^{2} - 4) (x^{2} + 12))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.29.6

Rewrite $4$ as $2^{2}$ .

$\frac{8 x ((x^{2} - 2^{2}) (x^{2} + 12))}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.5.29.7

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 2$ .

$\frac{8 x (x + 2) (x - 2) (x^{2} + 12)}{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.6

Combine terms.

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Step 2.1.2.11.6.1

Multiply the exponents in ${({(x + 2)}^{2})}^{2}$ .

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Step 2.1.2.11.6.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{8 x (x + 2) (x - 2) (x^{2} + 12)}{{(x + 2)}^{2 \cdot 2} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.6.1.2

Multiply $2$ by $2$ .

$\frac{8 x (x + 2) (x - 2) (x^{2} + 12)}{{(x + 2)}^{4} {({(x - 2)}^{2})}^{2}}$

Step 2.1.2.11.6.2

Multiply the exponents in ${({(x - 2)}^{2})}^{2}$ .

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Step 2.1.2.11.6.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{8 x (x + 2) (x - 2) (x^{2} + 12)}{{(x + 2)}^{4} {(x - 2)}^{2 \cdot 2}}$

Step 2.1.2.11.6.2.2

Multiply $2$ by $2$ .

$\frac{8 x (x + 2) (x - 2) (x^{2} + 12)}{{(x + 2)}^{4} {(x - 2)}^{4}}$

Step 2.1.2.11.6.3

Cancel the common factor of $x + 2$ and ${(x + 2)}^{4}$ .

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Step 2.1.2.11.6.3.1

Factor $x + 2$ out of $8 x (x + 2) (x - 2) (x^{2} + 12)$ .

$\frac{(x + 2) ((8 x (x - 2)) (x^{2} + 12))}{{(x + 2)}^{4} {(x - 2)}^{4}}$

Step 2.1.2.11.6.3.2

Cancel the common factors.

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Step 2.1.2.11.6.3.2.1

Factor $x + 2$ out of ${(x + 2)}^{4} {(x - 2)}^{4}$ .

$\frac{(x + 2) ((8 x (x - 2)) (x^{2} + 12))}{(x + 2) ({(x + 2)}^{3} {(x - 2)}^{4})}$

Step 2.1.2.11.6.3.2.2

Cancel the common factor.

$\frac{(x + 2) ((8 x (x - 2)) (x^{2} + 12))}{(x + 2) ({(x + 2)}^{3} {(x - 2)}^{4})}$

Step 2.1.2.11.6.3.2.3

Rewrite the expression.

$\frac{(8 x (x - 2)) (x^{2} + 12)}{{(x + 2)}^{3} {(x - 2)}^{4}}$

Step 2.1.2.11.6.4

Cancel the common factor of $x - 2$ and ${(x - 2)}^{4}$ .

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Step 2.1.2.11.6.4.1

Factor $x - 2$ out of $8 x (x - 2) (x^{2} + 12)$ .

$\frac{(x - 2) (8 x (x^{2} + 12))}{{(x + 2)}^{3} {(x - 2)}^{4}}$

Step 2.1.2.11.6.4.2

Cancel the common factors.

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Step 2.1.2.11.6.4.2.1

Factor $x - 2$ out of ${(x + 2)}^{3} {(x - 2)}^{4}$ .

$\frac{(x - 2) (8 x (x^{2} + 12))}{(x - 2) ({(x + 2)}^{3} {(x - 2)}^{3})}$

Step 2.1.2.11.6.4.2.2

Cancel the common factor.

$\frac{(x - 2) (8 x (x^{2} + 12))}{(x - 2) ({(x + 2)}^{3} {(x - 2)}^{3})}$

Step 2.1.2.11.6.4.2.3

Rewrite the expression.

$f'' (x) = \frac{8 x (x^{2} + 12)}{{(x + 2)}^{3} {(x - 2)}^{3}}$

Step 2.1.3

The second derivative of $f (x)$ with respect to $x$ is $\frac{8 x (x^{2} + 12)}{{(x + 2)}^{3} {(x - 2)}^{3}}$ .

$\frac{8 x (x^{2} + 12)}{{(x + 2)}^{3} {(x - 2)}^{3}}$

Step 2.2

Set the second derivative equal to $0$ then solve the equation $\frac{8 x (x^{2} + 12)}{{(x + 2)}^{3} {(x - 2)}^{3}} = 0$ .

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Step 2.2.1

Set the second derivative equal to $0$ .

$\frac{8 x (x^{2} + 12)}{{(x + 2)}^{3} {(x - 2)}^{3}} = 0$

Step 2.2.2

Set the numerator equal to zero.

$8 x (x^{2} + 12) = 0$

Step 2.2.3

Solve the equation for $x$ .

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Step 2.2.3.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$x^{2} + 12 = 0$

Step 2.2.3.2

Set $x$ equal to $0$ .

$x = 0$

Step 2.2.3.3

Set $x^{2} + 12$ equal to $0$ and solve for $x$ .

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Step 2.2.3.3.1

Set $x^{2} + 12$ equal to $0$ .

$x^{2} + 12 = 0$

Step 2.2.3.3.2

Solve $x^{2} + 12 = 0$ for $x$ .

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Step 2.2.3.3.2.1

Subtract $12$ from both sides of the equation.

$x^{2} = - 12$

Step 2.2.3.3.2.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{- 12}$

Step 2.2.3.3.2.3

Simplify $\pm \sqrt{- 12}$ .

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Step 2.2.3.3.2.3.1

Rewrite $- 12$ as $- 1 (12)$ .

$x = \pm \sqrt{- 1 (12)}$

Step 2.2.3.3.2.3.2

Rewrite $\sqrt{- 1 (12)}$ as $\sqrt{- 1} \cdot \sqrt{12}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{12}$

Step 2.2.3.3.2.3.3

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i \cdot \sqrt{12}$

Step 2.2.3.3.2.3.4

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 2.2.3.3.2.3.4.1

Factor $4$ out of $12$ .

$x = \pm i \cdot \sqrt{4 (3)}$

Step 2.2.3.3.2.3.4.2

Rewrite $4$ as $2^{2}$ .

$x = \pm i \cdot \sqrt{2^{2} \cdot 3}$

Step 2.2.3.3.2.3.5

Pull terms out from under the radical.

$x = \pm i \cdot (2 \sqrt{3})$

Step 2.2.3.3.2.3.6

Move $2$ to the left of $i$ .

$x = \pm 2 i \sqrt{3}$

Step 2.2.3.3.2.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.2.3.3.2.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 2 i \sqrt{3}$

Step 2.2.3.3.2.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 2 i \sqrt{3}$

Step 2.2.3.3.2.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 2 i \sqrt{3}, - 2 i \sqrt{3}$

Step 2.2.3.4

The final solution is all the values that make $8 x (x^{2} + 12) = 0$ true.

$x = 0, 2 i \sqrt{3}, - 2 i \sqrt{3}$

Step 3

Find the domain of $f (x) = \frac{x^{3}}{x^{2} - 4}$ .

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Step 3.1

Set the denominator in $\frac{x^{3}}{x^{2} - 4}$ equal to $0$ to find where the expression is undefined.

$x^{2} - 4 = 0$

Step 3.2

Solve for $x$ .

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Step 3.2.1

Add $4$ to both sides of the equation.

$x^{2} = 4$

Step 3.2.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{4}$

Step 3.2.3

Simplify $\pm \sqrt{4}$ .

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Step 3.2.3.1

Rewrite $4$ as $2^{2}$ .

$x = \pm \sqrt{2^{2}}$

Step 3.2.3.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 2$

Step 3.2.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.2.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 2$

Step 3.2.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 2$

Step 3.2.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 2, - 2$

Step 3.3

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(- \infty, - 2) \cup (- 2, 2) \cup (2, \infty)$

Set-Builder Notation:

${x | x \neq 2, - 2}$

Interval Notation:

$(- \infty, - 2) \cup (- 2, 2) \cup (2, \infty)$

Set-Builder Notation:

${x | x \neq 2, - 2}$

Step 4

Create intervals around the $x$ -values where the second derivative is zero or undefined.

$(- \infty, - 2) \cup (- 2, 0) \cup (0, 2) \cup (2, \infty)$

Step 5

Substitute any number from the interval $(- \infty, - 2)$ into the second derivative and evaluate to determine the concavity.

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Step 5.1

Replace the variable $x$ with $- 4$ in the expression.

$f'' (- 4) = \frac{8 (- 4) ({(- 4)}^{2} + 12)}{{((- 4) + 2)}^{3} {((- 4) - 2)}^{3}}$

Step 5.2

Simplify the result.

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Step 5.2.1

Multiply $8$ by $- 4$ .

$f'' (- 4) = \frac{- 32 ({(- 4)}^{2} + 12)}{{(- 4 + 2)}^{3} {(- 4 - 2)}^{3}}$

Step 5.2.2

Simplify the denominator.

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Step 5.2.2.1

Add $- 4$ and $2$ .

$f'' (- 4) = \frac{- 32 ({(- 4)}^{2} + 12)}{{(- 2)}^{3} {(- 4 - 2)}^{3}}$

Step 5.2.2.2

Subtract $2$ from $- 4$ .

$f'' (- 4) = \frac{- 32 ({(- 4)}^{2} + 12)}{{(- 2)}^{3} {(- 6)}^{3}}$

Step 5.2.2.3

Raise $- 2$ to the power of $3$ .

$f'' (- 4) = \frac{- 32 ({(- 4)}^{2} + 12)}{- 8 {(- 6)}^{3}}$

Step 5.2.2.4

Raise $- 6$ to the power of $3$ .

$f'' (- 4) = \frac{- 32 ({(- 4)}^{2} + 12)}{- 8 \cdot - 216}$

Step 5.2.3

Simplify the numerator.

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Step 5.2.3.1

Raise $- 4$ to the power of $2$ .

$f'' (- 4) = \frac{- 32 (16 + 12)}{- 8 \cdot - 216}$

Step 5.2.3.2

Add $16$ and $12$ .

$f'' (- 4) = \frac{- 32 \cdot 28}{- 8 \cdot - 216}$

Step 5.2.4

Reduce the expression by cancelling the common factors.

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Step 5.2.4.1

Multiply $- 8$ by $- 216$ .

$f'' (- 4) = \frac{- 32 \cdot 28}{1728}$

Step 5.2.4.2

Multiply $- 32$ by $28$ .

$f'' (- 4) = \frac{- 896}{1728}$

Step 5.2.4.3

Cancel the common factor of $- 896$ and $1728$ .

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Step 5.2.4.3.1

Factor $64$ out of $- 896$ .

$f'' (- 4) = \frac{64 (- 14)}{1728}$

Step 5.2.4.3.2

Cancel the common factors.

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Step 5.2.4.3.2.1

Factor $64$ out of $1728$ .

$f'' (- 4) = \frac{64 \cdot - 14}{64 \cdot 27}$

Step 5.2.4.3.2.2

Cancel the common factor.

$f'' (- 4) = \frac{64 \cdot - 14}{64 \cdot 27}$

Step 5.2.4.3.2.3

Rewrite the expression.

$f'' (- 4) = \frac{- 14}{27}$

Step 5.2.4.4

Move the negative in front of the fraction.

$f'' (- 4) = - \frac{14}{27}$

Step 5.2.5

The final answer is $- \frac{14}{27}$ .

$- \frac{14}{27}$

Step 5.3

The graph is concave down on the interval $(- \infty, - 2)$ because $f'' (- 4)$ is negative.

Concave down on $(- \infty, - 2)$ since $f'' (x)$ is negative

Step 6

Substitute any number from the interval $(- 2, 0)$ into the second derivative and evaluate to determine the concavity.

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Step 6.1

Replace the variable $x$ with $- 1$ in the expression.

$f'' (- 1) = \frac{8 (- 1) ({(- 1)}^{2} + 12)}{{((- 1) + 2)}^{3} {((- 1) - 2)}^{3}}$

Step 6.2

Simplify the result.

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Step 6.2.1

Multiply $8$ by $- 1$ .

$f'' (- 1) = \frac{- 8 ({(- 1)}^{2} + 12)}{{(- 1 + 2)}^{3} {(- 1 - 2)}^{3}}$

Step 6.2.2

Simplify the denominator.

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Step 6.2.2.1

Add $- 1$ and $2$ .

$f'' (- 1) = \frac{- 8 ({(- 1)}^{2} + 12)}{1^{3} {(- 1 - 2)}^{3}}$

Step 6.2.2.2

Subtract $2$ from $- 1$ .

$f'' (- 1) = \frac{- 8 ({(- 1)}^{2} + 12)}{1^{3} {(- 3)}^{3}}$

Step 6.2.2.3

One to any power is one.

$f'' (- 1) = \frac{- 8 ({(- 1)}^{2} + 12)}{1 {(- 3)}^{3}}$

Step 6.2.2.4

Raise $- 3$ to the power of $3$ .

$f'' (- 1) = \frac{- 8 ({(- 1)}^{2} + 12)}{1 \cdot - 27}$

Step 6.2.2.5

Multiply $- 27$ by $1$ .

$f'' (- 1) = \frac{- 8 ({(- 1)}^{2} + 12)}{- 27}$

Step 6.2.3

Simplify the numerator.

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Step 6.2.3.1

Raise $- 1$ to the power of $2$ .

$f'' (- 1) = \frac{- 8 (1 + 12)}{- 27}$

Step 6.2.3.2

Add $1$ and $12$ .

$f'' (- 1) = \frac{- 8 \cdot 13}{- 27}$

Step 6.2.4

Reduce the expression by cancelling the common factors.

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Step 6.2.4.1

Multiply $- 8$ by $13$ .

$f'' (- 1) = \frac{- 104}{- 27}$

Step 6.2.4.2

Dividing two negative values results in a positive value.

$f'' (- 1) = \frac{104}{27}$

Step 6.2.5

The final answer is $\frac{104}{27}$ .

$\frac{104}{27}$

Step 6.3

The graph is concave up on the interval $(- 2, 0)$ because $f'' (- 1)$ is positive.

Concave up on $(- 2, 0)$ since $f'' (x)$ is positive

Step 7

Substitute any number from the interval $(0, 2)$ into the second derivative and evaluate to determine the concavity.

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Step 7.1

Replace the variable $x$ with $1$ in the expression.

$f'' (1) = \frac{8 (1) ({(1)}^{2} + 12)}{{((1) + 2)}^{3} {((1) - 2)}^{3}}$

Step 7.2

Simplify the result.

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Step 7.2.1

Multiply $8$ by $1$ .

$f'' (1) = \frac{8 (1^{2} + 12)}{{(1 + 2)}^{3} {(1 - 2)}^{3}}$

Step 7.2.2

Simplify the denominator.

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Step 7.2.2.1

Add $1$ and $2$ .

$f'' (1) = \frac{8 (1^{2} + 12)}{3^{3} {(1 - 2)}^{3}}$

Step 7.2.2.2

Subtract $2$ from $1$ .

$f'' (1) = \frac{8 (1^{2} + 12)}{3^{3} {(- 1)}^{3}}$

Step 7.2.2.3

Raise $3$ to the power of $3$ .

$f'' (1) = \frac{8 (1^{2} + 12)}{27 {(- 1)}^{3}}$

Step 7.2.2.4

Raise $- 1$ to the power of $3$ .

$f'' (1) = \frac{8 (1^{2} + 12)}{27 \cdot - 1}$

Step 7.2.3

Simplify the numerator.

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Step 7.2.3.1

One to any power is one.

$f'' (1) = \frac{8 (1 + 12)}{27 \cdot - 1}$

Step 7.2.3.2

Add $1$ and $12$ .

$f'' (1) = \frac{8 \cdot 13}{27 \cdot - 1}$

Step 7.2.4

Simplify the expression.

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Step 7.2.4.1

Multiply $27$ by $- 1$ .

$f'' (1) = \frac{8 \cdot 13}{- 27}$

Step 7.2.4.2

Multiply $8$ by $13$ .

$f'' (1) = \frac{104}{- 27}$

Step 7.2.4.3

Move the negative in front of the fraction.

$f'' (1) = - \frac{104}{27}$

Step 7.2.5

The final answer is $- \frac{104}{27}$ .

$- \frac{104}{27}$

Step 7.3

The graph is concave down on the interval $(0, 2)$ because $f'' (1)$ is negative.

Concave down on $(0, 2)$ since $f'' (x)$ is negative

Step 8

Substitute any number from the interval $(2, \infty)$ into the second derivative and evaluate to determine the concavity.

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Step 8.1

Replace the variable $x$ with $4$ in the expression.

$f'' (4) = \frac{8 (4) ({(4)}^{2} + 12)}{{((4) + 2)}^{3} {((4) - 2)}^{3}}$

Step 8.2

Simplify the result.

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Step 8.2.1

Multiply $8$ by $4$ .

$f'' (4) = \frac{32 (4^{2} + 12)}{{(4 + 2)}^{3} {(4 - 2)}^{3}}$

Step 8.2.2

Simplify the denominator.

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Step 8.2.2.1

Add $4$ and $2$ .

$f'' (4) = \frac{32 (4^{2} + 12)}{6^{3} {(4 - 2)}^{3}}$

Step 8.2.2.2

Subtract $2$ from $4$ .

$f'' (4) = \frac{32 (4^{2} + 12)}{6^{3} \cdot 2^{3}}$

Step 8.2.2.3

Raise $6$ to the power of $3$ .

$f'' (4) = \frac{32 (4^{2} + 12)}{216 \cdot 2^{3}}$

Step 8.2.2.4

Raise $2$ to the power of $3$ .

$f'' (4) = \frac{32 (4^{2} + 12)}{216 \cdot 8}$

Step 8.2.3

Simplify the numerator.

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Step 8.2.3.1

Raise $4$ to the power of $2$ .

$f'' (4) = \frac{32 (16 + 12)}{216 \cdot 8}$

Step 8.2.3.2

Add $16$ and $12$ .

$f'' (4) = \frac{32 \cdot 28}{216 \cdot 8}$

Step 8.2.4

Reduce the expression by cancelling the common factors.

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Step 8.2.4.1

Multiply $216$ by $8$ .

$f'' (4) = \frac{32 \cdot 28}{1728}$

Step 8.2.4.2

Multiply $32$ by $28$ .

$f'' (4) = \frac{896}{1728}$

Step 8.2.4.3

Cancel the common factor of $896$ and $1728$ .

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Step 8.2.4.3.1

Factor $64$ out of $896$ .

$f'' (4) = \frac{64 (14)}{1728}$

Step 8.2.4.3.2

Cancel the common factors.

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Step 8.2.4.3.2.1

Factor $64$ out of $1728$ .

$f'' (4) = \frac{64 \cdot 14}{64 \cdot 27}$

Step 8.2.4.3.2.2

Cancel the common factor.

$f'' (4) = \frac{64 \cdot 14}{64 \cdot 27}$

Step 8.2.4.3.2.3

Rewrite the expression.

$f'' (4) = \frac{14}{27}$

Step 8.2.5

The final answer is $\frac{14}{27}$ .

$\frac{14}{27}$

Step 8.3

The graph is concave up on the interval $(2, \infty)$ because $f'' (4)$ is positive.

Concave up on $(2, \infty)$ since $f'' (x)$ is positive

Step 9

The graph is concave down when the second derivative is negative and concave up when the second derivative is positive.

Concave down on $(- \infty, - 2)$ since $f'' (x)$ is negative

Concave up on $(- 2, 0)$ since $f'' (x)$ is positive

Concave down on $(0, 2)$ since $f'' (x)$ is negative

Concave up on $(2, \infty)$ since $f'' (x)$ is positive

Step 10