Calculus Examples

$y = \frac{{(x + 1)}^{2}}{(x + 2) (x + 3)}$

Step 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 $f (x) = {(x + 1)}^{2}$ and $g (x) = (x + 2) (x + 3)$ .

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

Step 2

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 $g (x) = x + 1$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $x + 1$ .

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u$ with $x + 1$ .

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

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $x + 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [1]$ .

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

Step 3.2

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

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

Step 3.3

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

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

Step 3.4

Simplify the expression.

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Step 3.4.1

Add $1$ and $0$ .

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

Step 3.4.2

Multiply $2$ by $1$ .

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

Step 4

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 $g (x) = x + 3$ .

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

Step 5

Differentiate.

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Step 5.1

By the Sum Rule, the derivative of $x + 3$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [3]$ .

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

Step 5.2

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

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

Step 5.3

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

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

Step 5.4

Simplify the expression.

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Step 5.4.1

Add $1$ and $0$ .

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

Step 5.4.2

Multiply $x + 2$ by $1$ .

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

Step 5.5

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

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

Step 5.6

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

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

Step 5.7

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

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

Step 5.8

Simplify by adding terms.

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Step 5.8.1

Add $1$ and $0$ .

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

Step 5.8.2

Multiply $x + 3$ by $1$ .

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

Step 5.8.3

Add $x$ and $x$ .

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

Step 5.8.4

Add $2$ and $3$ .

$\frac{(x + 2) (x + 3) (2 (x + 1)) - {(x + 1)}^{2} (2 x + 5)}{{((x + 2) (x + 3))}^{2}}$

Step 6

Simplify.

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Step 6.1

Apply the product rule to $(x + 2) (x + 3)$ .

$\frac{(x + 2) (x + 3) (2 (x + 1)) - {(x + 1)}^{2} (2 x + 5)}{{(x + 2)}^{2} {(x + 3)}^{2}}$

Step 6.2

Apply the distributive property.

$\frac{(x + 2) (x + 3) (2 x + 2 \cdot 1) - {(x + 1)}^{2} (2 x + 5)}{{(x + 2)}^{2} {(x + 3)}^{2}}$

Step 6.3

Simplify the numerator.

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Step 6.3.1

Simplify each term.

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Step 6.3.1.1

Expand $(x + 2) (x + 3)$ using the FOIL Method.

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Step 6.3.1.1.1

Apply the distributive property.

$\frac{(x (x + 3) + 2 (x + 3)) (2 x + 2 \cdot 1) - {(x + 1)}^{2} (2 x + 5)}{{(x + 2)}^{2} {(x + 3)}^{2}}$

Step 6.3.1.1.2

Apply the distributive property.

$\frac{(x \cdot x + x \cdot 3 + 2 (x + 3)) (2 x + 2 \cdot 1) - {(x + 1)}^{2} (2 x + 5)}{{(x + 2)}^{2} {(x + 3)}^{2}}$

Step 6.3.1.1.3

Apply the distributive property.

$\frac{(x \cdot x + x \cdot 3 + 2 x + 2 \cdot 3) (2 x + 2 \cdot 1) - {(x + 1)}^{2} (2 x + 5)}{{(x + 2)}^{2} {(x + 3)}^{2}}$

Step 6.3.1.2

Simplify and combine like terms.

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Step 6.3.1.2.1

Simplify each term.

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Step 6.3.1.2.1.1

Multiply $x$ by $x$ .

$\frac{(x^{2} + x \cdot 3 + 2 x + 2 \cdot 3) (2 x + 2 \cdot 1) - {(x + 1)}^{2} (2 x + 5)}{{(x + 2)}^{2} {(x + 3)}^{2}}$

Step 6.3.1.2.1.2

Move $3$ to the left of $x$ .

$\frac{(x^{2} + 3 \cdot x + 2 x + 2 \cdot 3) (2 x + 2 \cdot 1) - {(x + 1)}^{2} (2 x + 5)}{{(x + 2)}^{2} {(x + 3)}^{2}}$

Step 6.3.1.2.1.3

Multiply $2$ by $3$ .

$\frac{(x^{2} + 3 x + 2 x + 6) (2 x + 2 \cdot 1) - {(x + 1)}^{2} (2 x + 5)}{{(x + 2)}^{2} {(x + 3)}^{2}}$

Step 6.3.1.2.2

Add $3 x$ and $2 x$ .

$\frac{(x^{2} + 5 x + 6) (2 x + 2 \cdot 1) - {(x + 1)}^{2} (2 x + 5)}{{(x + 2)}^{2} {(x + 3)}^{2}}$

Step 6.3.1.3

Multiply $2$ by $1$ .

$\frac{(x^{2} + 5 x + 6) (2 x + 2) - {(x + 1)}^{2} (2 x + 5)}{{(x + 2)}^{2} {(x + 3)}^{2}}$

Step 6.3.1.4

Expand $(x^{2} + 5 x + 6) (2 x + 2)$ by multiplying each term in the first expression by each term in the second expression.

$\frac{x^{2} (2 x) + x^{2} \cdot 2 + 5 x (2 x) + 5 x \cdot 2 + 6 (2 x) + 6 \cdot 2 - {(x + 1)}^{2} (2 x + 5)}{{(x + 2)}^{2} {(x + 3)}^{2}}$

Step 6.3.1.5

Simplify each term.

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Step 6.3.1.5.1

Rewrite using the commutative property of multiplication.

$\frac{2 x^{2} x + x^{2} \cdot 2 + 5 x (2 x) + 5 x \cdot 2 + 6 (2 x) + 6 \cdot 2 - {(x + 1)}^{2} (2 x + 5)}{{(x + 2)}^{2} {(x + 3)}^{2}}$

Step 6.3.1.5.2

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

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Step 6.3.1.5.2.1

Move $x$ .

$\frac{2 (x \cdot x^{2}) + x^{2} \cdot 2 + 5 x (2 x) + 5 x \cdot 2 + 6 (2 x) + 6 \cdot 2 - {(x + 1)}^{2} (2 x + 5)}{{(x + 2)}^{2} {(x + 3)}^{2}}$

Step 6.3.1.5.2.2

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

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Step 6.3.1.5.2.2.1

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

$\frac{2 (x^{1} x^{2}) + x^{2} \cdot 2 + 5 x (2 x) + 5 x \cdot 2 + 6 (2 x) + 6 \cdot 2 - {(x + 1)}^{2} (2 x + 5)}{{(x + 2)}^{2} {(x + 3)}^{2}}$

Step 6.3.1.5.2.2.2

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

$\frac{2 x^{1 + 2} + x^{2} \cdot 2 + 5 x (2 x) + 5 x \cdot 2 + 6 (2 x) + 6 \cdot 2 - {(x + 1)}^{2} (2 x + 5)}{{(x + 2)}^{2} {(x + 3)}^{2}}$

Step 6.3.1.5.2.3

Add $1$ and $2$ .

$\frac{2 x^{3} + x^{2} \cdot 2 + 5 x (2 x) + 5 x \cdot 2 + 6 (2 x) + 6 \cdot 2 - {(x + 1)}^{2} (2 x + 5)}{{(x + 2)}^{2} {(x + 3)}^{2}}$

Step 6.3.1.5.3

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

$\frac{2 x^{3} + 2 \cdot x^{2} + 5 x (2 x) + 5 x \cdot 2 + 6 (2 x) + 6 \cdot 2 - {(x + 1)}^{2} (2 x + 5)}{{(x + 2)}^{2} {(x + 3)}^{2}}$

Step 6.3.1.5.4

Rewrite using the commutative property of multiplication.

$\frac{2 x^{3} + 2 x^{2} + 5 \cdot 2 x \cdot x + 5 x \cdot 2 + 6 (2 x) + 6 \cdot 2 - {(x + 1)}^{2} (2 x + 5)}{{(x + 2)}^{2} {(x + 3)}^{2}}$

Step 6.3.1.5.5

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

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Step 6.3.1.5.5.1

Move $x$ .

$\frac{2 x^{3} + 2 x^{2} + 5 \cdot 2 (x \cdot x) + 5 x \cdot 2 + 6 (2 x) + 6 \cdot 2 - {(x + 1)}^{2} (2 x + 5)}{{(x + 2)}^{2} {(x + 3)}^{2}}$

Step 6.3.1.5.5.2

Multiply $x$ by $x$ .

$\frac{2 x^{3} + 2 x^{2} + 5 \cdot 2 x^{2} + 5 x \cdot 2 + 6 (2 x) + 6 \cdot 2 - {(x + 1)}^{2} (2 x + 5)}{{(x + 2)}^{2} {(x + 3)}^{2}}$

Step 6.3.1.5.6

Multiply $5$ by $2$ .

$\frac{2 x^{3} + 2 x^{2} + 10 x^{2} + 5 x \cdot 2 + 6 (2 x) + 6 \cdot 2 - {(x + 1)}^{2} (2 x + 5)}{{(x + 2)}^{2} {(x + 3)}^{2}}$

Step 6.3.1.5.7

Multiply $2$ by $5$ .

$\frac{2 x^{3} + 2 x^{2} + 10 x^{2} + 10 x + 6 (2 x) + 6 \cdot 2 - {(x + 1)}^{2} (2 x + 5)}{{(x + 2)}^{2} {(x + 3)}^{2}}$

Step 6.3.1.5.8

Multiply $2$ by $6$ .

$\frac{2 x^{3} + 2 x^{2} + 10 x^{2} + 10 x + 12 x + 6 \cdot 2 - {(x + 1)}^{2} (2 x + 5)}{{(x + 2)}^{2} {(x + 3)}^{2}}$

Step 6.3.1.5.9

Multiply $6$ by $2$ .

$\frac{2 x^{3} + 2 x^{2} + 10 x^{2} + 10 x + 12 x + 12 - {(x + 1)}^{2} (2 x + 5)}{{(x + 2)}^{2} {(x + 3)}^{2}}$

Step 6.3.1.6

Add $2 x^{2}$ and $10 x^{2}$ .

$\frac{2 x^{3} + 12 x^{2} + 10 x + 12 x + 12 - {(x + 1)}^{2} (2 x + 5)}{{(x + 2)}^{2} {(x + 3)}^{2}}$

Step 6.3.1.7

Add $10 x$ and $12 x$ .

$\frac{2 x^{3} + 12 x^{2} + 22 x + 12 - {(x + 1)}^{2} (2 x + 5)}{{(x + 2)}^{2} {(x + 3)}^{2}}$

Step 6.3.1.8

Rewrite ${(x + 1)}^{2}$ as $(x + 1) (x + 1)$ .

$\frac{2 x^{3} + 12 x^{2} + 22 x + 12 - ((x + 1) (x + 1)) (2 x + 5)}{{(x + 2)}^{2} {(x + 3)}^{2}}$

Step 6.3.1.9

Expand $(x + 1) (x + 1)$ using the FOIL Method.

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Step 6.3.1.9.1

Apply the distributive property.

$\frac{2 x^{3} + 12 x^{2} + 22 x + 12 - (x (x + 1) + 1 (x + 1)) (2 x + 5)}{{(x + 2)}^{2} {(x + 3)}^{2}}$

Step 6.3.1.9.2

Apply the distributive property.

$\frac{2 x^{3} + 12 x^{2} + 22 x + 12 - (x \cdot x + x \cdot 1 + 1 (x + 1)) (2 x + 5)}{{(x + 2)}^{2} {(x + 3)}^{2}}$

Step 6.3.1.9.3

Apply the distributive property.

$\frac{2 x^{3} + 12 x^{2} + 22 x + 12 - (x \cdot x + x \cdot 1 + 1 x + 1 \cdot 1) (2 x + 5)}{{(x + 2)}^{2} {(x + 3)}^{2}}$

Step 6.3.1.10

Simplify and combine like terms.

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Step 6.3.1.10.1

Simplify each term.

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Step 6.3.1.10.1.1

Multiply $x$ by $x$ .

$\frac{2 x^{3} + 12 x^{2} + 22 x + 12 - (x^{2} + x \cdot 1 + 1 x + 1 \cdot 1) (2 x + 5)}{{(x + 2)}^{2} {(x + 3)}^{2}}$

Step 6.3.1.10.1.2

Multiply $x$ by $1$ .

$\frac{2 x^{3} + 12 x^{2} + 22 x + 12 - (x^{2} + x + 1 x + 1 \cdot 1) (2 x + 5)}{{(x + 2)}^{2} {(x + 3)}^{2}}$

Step 6.3.1.10.1.3

Multiply $x$ by $1$ .

$\frac{2 x^{3} + 12 x^{2} + 22 x + 12 - (x^{2} + x + x + 1 \cdot 1) (2 x + 5)}{{(x + 2)}^{2} {(x + 3)}^{2}}$

Step 6.3.1.10.1.4

Multiply $1$ by $1$ .

$\frac{2 x^{3} + 12 x^{2} + 22 x + 12 - (x^{2} + x + x + 1) (2 x + 5)}{{(x + 2)}^{2} {(x + 3)}^{2}}$

Step 6.3.1.10.2

Add $x$ and $x$ .

$\frac{2 x^{3} + 12 x^{2} + 22 x + 12 - (x^{2} + 2 x + 1) (2 x + 5)}{{(x + 2)}^{2} {(x + 3)}^{2}}$

Step 6.3.1.11

Apply the distributive property.

$\frac{2 x^{3} + 12 x^{2} + 22 x + 12 + (- x^{2} - (2 x) - 1 \cdot 1) (2 x + 5)}{{(x + 2)}^{2} {(x + 3)}^{2}}$

Step 6.3.1.12

Simplify.

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Step 6.3.1.12.1

Multiply $2$ by $- 1$ .

$\frac{2 x^{3} + 12 x^{2} + 22 x + 12 + (- x^{2} - 2 x - 1 \cdot 1) (2 x + 5)}{{(x + 2)}^{2} {(x + 3)}^{2}}$

Step 6.3.1.12.2

Multiply $- 1$ by $1$ .

$\frac{2 x^{3} + 12 x^{2} + 22 x + 12 + (- x^{2} - 2 x - 1) (2 x + 5)}{{(x + 2)}^{2} {(x + 3)}^{2}}$

Step 6.3.1.13

Expand $(- x^{2} - 2 x - 1) (2 x + 5)$ by multiplying each term in the first expression by each term in the second expression.

$\frac{2 x^{3} + 12 x^{2} + 22 x + 12 - x^{2} (2 x) - x^{2} \cdot 5 - 2 x (2 x) - 2 x \cdot 5 - 1 (2 x) - 1 \cdot 5}{{(x + 2)}^{2} {(x + 3)}^{2}}$

Step 6.3.1.14

Simplify each term.

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Step 6.3.1.14.1

Rewrite using the commutative property of multiplication.

$\frac{2 x^{3} + 12 x^{2} + 22 x + 12 - 1 \cdot 2 x^{2} x - x^{2} \cdot 5 - 2 x (2 x) - 2 x \cdot 5 - 1 (2 x) - 1 \cdot 5}{{(x + 2)}^{2} {(x + 3)}^{2}}$

Step 6.3.1.14.2

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

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Step 6.3.1.14.2.1

Move $x$ .

$\frac{2 x^{3} + 12 x^{2} + 22 x + 12 - 1 \cdot 2 (x \cdot x^{2}) - x^{2} \cdot 5 - 2 x (2 x) - 2 x \cdot 5 - 1 (2 x) - 1 \cdot 5}{{(x + 2)}^{2} {(x + 3)}^{2}}$

Step 6.3.1.14.2.2

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

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Step 6.3.1.14.2.2.1

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

$\frac{2 x^{3} + 12 x^{2} + 22 x + 12 - 1 \cdot 2 (x^{1} x^{2}) - x^{2} \cdot 5 - 2 x (2 x) - 2 x \cdot 5 - 1 (2 x) - 1 \cdot 5}{{(x + 2)}^{2} {(x + 3)}^{2}}$

Step 6.3.1.14.2.2.2

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

$\frac{2 x^{3} + 12 x^{2} + 22 x + 12 - 1 \cdot 2 x^{1 + 2} - x^{2} \cdot 5 - 2 x (2 x) - 2 x \cdot 5 - 1 (2 x) - 1 \cdot 5}{{(x + 2)}^{2} {(x + 3)}^{2}}$

Step 6.3.1.14.2.3

Add $1$ and $2$ .

$\frac{2 x^{3} + 12 x^{2} + 22 x + 12 - 1 \cdot 2 x^{3} - x^{2} \cdot 5 - 2 x (2 x) - 2 x \cdot 5 - 1 (2 x) - 1 \cdot 5}{{(x + 2)}^{2} {(x + 3)}^{2}}$

Step 6.3.1.14.3

Multiply $- 1$ by $2$ .

$\frac{2 x^{3} + 12 x^{2} + 22 x + 12 - 2 x^{3} - x^{2} \cdot 5 - 2 x (2 x) - 2 x \cdot 5 - 1 (2 x) - 1 \cdot 5}{{(x + 2)}^{2} {(x + 3)}^{2}}$

Step 6.3.1.14.4

Multiply $5$ by $- 1$ .

$\frac{2 x^{3} + 12 x^{2} + 22 x + 12 - 2 x^{3} - 5 x^{2} - 2 x (2 x) - 2 x \cdot 5 - 1 (2 x) - 1 \cdot 5}{{(x + 2)}^{2} {(x + 3)}^{2}}$

Step 6.3.1.14.5

Rewrite using the commutative property of multiplication.

$\frac{2 x^{3} + 12 x^{2} + 22 x + 12 - 2 x^{3} - 5 x^{2} - 2 \cdot 2 x \cdot x - 2 x \cdot 5 - 1 (2 x) - 1 \cdot 5}{{(x + 2)}^{2} {(x + 3)}^{2}}$

Step 6.3.1.14.6

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

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Step 6.3.1.14.6.1

Move $x$ .

$\frac{2 x^{3} + 12 x^{2} + 22 x + 12 - 2 x^{3} - 5 x^{2} - 2 \cdot 2 (x \cdot x) - 2 x \cdot 5 - 1 (2 x) - 1 \cdot 5}{{(x + 2)}^{2} {(x + 3)}^{2}}$

Step 6.3.1.14.6.2

Multiply $x$ by $x$ .

$\frac{2 x^{3} + 12 x^{2} + 22 x + 12 - 2 x^{3} - 5 x^{2} - 2 \cdot 2 x^{2} - 2 x \cdot 5 - 1 (2 x) - 1 \cdot 5}{{(x + 2)}^{2} {(x + 3)}^{2}}$

Step 6.3.1.14.7

Multiply $- 2$ by $2$ .

$\frac{2 x^{3} + 12 x^{2} + 22 x + 12 - 2 x^{3} - 5 x^{2} - 4 x^{2} - 2 x \cdot 5 - 1 (2 x) - 1 \cdot 5}{{(x + 2)}^{2} {(x + 3)}^{2}}$

Step 6.3.1.14.8

Multiply $5$ by $- 2$ .

$\frac{2 x^{3} + 12 x^{2} + 22 x + 12 - 2 x^{3} - 5 x^{2} - 4 x^{2} - 10 x - 1 (2 x) - 1 \cdot 5}{{(x + 2)}^{2} {(x + 3)}^{2}}$

Step 6.3.1.14.9

Multiply $2$ by $- 1$ .

$\frac{2 x^{3} + 12 x^{2} + 22 x + 12 - 2 x^{3} - 5 x^{2} - 4 x^{2} - 10 x - 2 x - 1 \cdot 5}{{(x + 2)}^{2} {(x + 3)}^{2}}$

Step 6.3.1.14.10

Multiply $- 1$ by $5$ .

$\frac{2 x^{3} + 12 x^{2} + 22 x + 12 - 2 x^{3} - 5 x^{2} - 4 x^{2} - 10 x - 2 x - 5}{{(x + 2)}^{2} {(x + 3)}^{2}}$

Step 6.3.1.15

Subtract $4 x^{2}$ from $- 5 x^{2}$ .

$\frac{2 x^{3} + 12 x^{2} + 22 x + 12 - 2 x^{3} - 9 x^{2} - 10 x - 2 x - 5}{{(x + 2)}^{2} {(x + 3)}^{2}}$

Step 6.3.1.16

Subtract $2 x$ from $- 10 x$ .

$\frac{2 x^{3} + 12 x^{2} + 22 x + 12 - 2 x^{3} - 9 x^{2} - 12 x - 5}{{(x + 2)}^{2} {(x + 3)}^{2}}$

Step 6.3.2

Combine the opposite terms in $2 x^{3} + 12 x^{2} + 22 x + 12 - 2 x^{3} - 9 x^{2} - 12 x - 5$ .

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Step 6.3.2.1

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

$\frac{12 x^{2} + 22 x + 12 + 0 - 9 x^{2} - 12 x - 5}{{(x + 2)}^{2} {(x + 3)}^{2}}$

Step 6.3.2.2

Add $12 x^{2} + 22 x + 12$ and $0$ .

$\frac{12 x^{2} + 22 x + 12 - 9 x^{2} - 12 x - 5}{{(x + 2)}^{2} {(x + 3)}^{2}}$

Step 6.3.3

Subtract $9 x^{2}$ from $12 x^{2}$ .

$\frac{3 x^{2} + 22 x + 12 - 12 x - 5}{{(x + 2)}^{2} {(x + 3)}^{2}}$

Step 6.3.4

Subtract $12 x$ from $22 x$ .

$\frac{3 x^{2} + 10 x + 12 - 5}{{(x + 2)}^{2} {(x + 3)}^{2}}$

Step 6.3.5

Subtract $5$ from $12$ .

$\frac{3 x^{2} + 10 x + 7}{{(x + 2)}^{2} {(x + 3)}^{2}}$

Step 6.4

Factor by grouping.

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Step 6.4.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot 7 = 21$ and whose sum is $b = 10$ .

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Step 6.4.1.1

Factor $10$ out of $10 x$ .

$\frac{3 x^{2} + 10 (x) + 7}{{(x + 2)}^{2} {(x + 3)}^{2}}$

Step 6.4.1.2

Rewrite $10$ as $3$ plus $7$

$\frac{3 x^{2} + (3 + 7) x + 7}{{(x + 2)}^{2} {(x + 3)}^{2}}$

Step 6.4.1.3

Apply the distributive property.

$\frac{3 x^{2} + 3 x + 7 x + 7}{{(x + 2)}^{2} {(x + 3)}^{2}}$

Step 6.4.2

Factor out the greatest common factor from each group.

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Step 6.4.2.1

Group the first two terms and the last two terms.

$\frac{(3 x^{2} + 3 x) + 7 x + 7}{{(x + 2)}^{2} {(x + 3)}^{2}}$

Step 6.4.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{3 x (x + 1) + 7 (x + 1)}{{(x + 2)}^{2} {(x + 3)}^{2}}$

Step 6.4.3

Factor the polynomial by factoring out the greatest common factor, $x + 1$ .

$\frac{(x + 1) (3 x + 7)}{{(x + 2)}^{2} {(x + 3)}^{2}}$