Calculus Examples

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

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 $f (x) = (x + 3) (x + 2)$ and $g (x) = (x - 3) (x - 2)$ .

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

Step 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 $f (x) = x + 3$ and $g (x) = x + 2$ .

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Simplify the expression.

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Step 3.4.1

Add $1$ and $0$ .

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

Step 3.4.2

Multiply $x + 3$ by $1$ .

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

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

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

Step 3.8

Simplify by adding terms.

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Step 3.8.1

Add $1$ and $0$ .

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

Step 3.8.2

Multiply $x + 2$ by $1$ .

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

Step 3.8.3

Add $x$ and $x$ .

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

Step 3.8.4

Add $3$ and $2$ .

$\frac{(x - 3) (x - 2) (2 x + 5) - (x + 3) (x + 2) \frac{d}{d x} [(x - 3) (x - 2)]}{{((x - 3) (x - 2))}^{2}}$

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 $f (x) = x - 3$ and $g (x) = x - 2$ .

$\frac{(x - 3) (x - 2) (2 x + 5) - (x + 3) (x + 2) ((x - 3) \frac{d}{d x} [x - 2] + (x - 2) \frac{d}{d x} [x - 3])}{{((x - 3) (x - 2))}^{2}}$

Step 5

Differentiate.

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Step 5.1

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

$\frac{(x - 3) (x - 2) (2 x + 5) - (x + 3) (x + 2) ((x - 3) (\frac{d}{d x} [x] + \frac{d}{d x} [- 2]) + (x - 2) \frac{d}{d x} [x - 3])}{{((x - 3) (x - 2))}^{2}}$

Step 5.2

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

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

Step 5.3

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

$\frac{(x - 3) (x - 2) (2 x + 5) - (x + 3) (x + 2) ((x - 3) (1 + 0) + (x - 2) \frac{d}{d x} [x - 3])}{{((x - 3) (x - 2))}^{2}}$

Step 5.4

Simplify the expression.

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Step 5.4.1

Add $1$ and $0$ .

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

Step 5.4.2

Multiply $x - 3$ by $1$ .

$\frac{(x - 3) (x - 2) (2 x + 5) - (x + 3) (x + 2) (x - 3 + (x - 2) \frac{d}{d x} [x - 3])}{{((x - 3) (x - 2))}^{2}}$

Step 5.5

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

$\frac{(x - 3) (x - 2) (2 x + 5) - (x + 3) (x + 2) (x - 3 + (x - 2) (\frac{d}{d x} [x] + \frac{d}{d x} [- 3]))}{{((x - 3) (x - 2))}^{2}}$

Step 5.6

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

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

Step 5.7

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

$\frac{(x - 3) (x - 2) (2 x + 5) - (x + 3) (x + 2) (x - 3 + (x - 2) (1 + 0))}{{((x - 3) (x - 2))}^{2}}$

Step 5.8

Simplify by adding terms.

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Step 5.8.1

Add $1$ and $0$ .

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

Step 5.8.2

Multiply $x - 2$ by $1$ .

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

Step 5.8.3

Add $x$ and $x$ .

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

Step 5.8.4

Subtract $2$ from $- 3$ .

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

Step 6

Simplify.

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Step 6.1

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

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

Step 6.2

Apply the distributive property.

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

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 $(x - 3) (x - 2)$ using the FOIL Method.

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Step 6.3.1.1.1

Apply the distributive property.

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

Step 6.3.1.1.2

Apply the distributive property.

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

Step 6.3.1.1.3

Apply the distributive property.

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

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 $x$ .

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

Step 6.3.1.2.1.2

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

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

Step 6.3.1.2.1.3

Multiply $- 3$ by $- 2$ .

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

Step 6.3.1.2.2

Subtract $3 x$ from $- 2 x$ .

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

Step 6.3.1.3

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

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

Step 6.3.1.4

Simplify each term.

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Step 6.3.1.4.1

Rewrite using the commutative property of multiplication.

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

Step 6.3.1.4.2

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

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Step 6.3.1.4.2.1

Move $x$ .

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

Step 6.3.1.4.2.2

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

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Step 6.3.1.4.2.2.1

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

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

Step 6.3.1.4.2.2.2

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

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

Step 6.3.1.4.2.3

Add $1$ and $2$ .

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

Step 6.3.1.4.3

Move $5$ to the left of $x^{2}$ .

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

Step 6.3.1.4.4

Rewrite using the commutative property of multiplication.

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

Step 6.3.1.4.5

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

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Step 6.3.1.4.5.1

Move $x$ .

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

Step 6.3.1.4.5.2

Multiply $x$ by $x$ .

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

Step 6.3.1.4.6

Multiply $- 5$ by $2$ .

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

Step 6.3.1.4.7

Multiply $5$ by $- 5$ .

$\frac{2 x^{3} + 5 x^{2} - 10 x^{2} - 25 x + 6 (2 x) + 6 \cdot 5 + (- x - 1 \cdot 3) (x + 2) (2 x - 5)}{{(x - 3)}^{2} {(x - 2)}^{2}}$

Step 6.3.1.4.8

Multiply $2$ by $6$ .

$\frac{2 x^{3} + 5 x^{2} - 10 x^{2} - 25 x + 12 x + 6 \cdot 5 + (- x - 1 \cdot 3) (x + 2) (2 x - 5)}{{(x - 3)}^{2} {(x - 2)}^{2}}$

Step 6.3.1.4.9

Multiply $6$ by $5$ .

$\frac{2 x^{3} + 5 x^{2} - 10 x^{2} - 25 x + 12 x + 30 + (- x - 1 \cdot 3) (x + 2) (2 x - 5)}{{(x - 3)}^{2} {(x - 2)}^{2}}$

Step 6.3.1.5

Subtract $10 x^{2}$ from $5 x^{2}$ .

$\frac{2 x^{3} - 5 x^{2} - 25 x + 12 x + 30 + (- x - 1 \cdot 3) (x + 2) (2 x - 5)}{{(x - 3)}^{2} {(x - 2)}^{2}}$

Step 6.3.1.6

Add $- 25 x$ and $12 x$ .

$\frac{2 x^{3} - 5 x^{2} - 13 x + 30 + (- x - 1 \cdot 3) (x + 2) (2 x - 5)}{{(x - 3)}^{2} {(x - 2)}^{2}}$

Step 6.3.1.7

Multiply $- 1$ by $3$ .

$\frac{2 x^{3} - 5 x^{2} - 13 x + 30 + (- x - 3) (x + 2) (2 x - 5)}{{(x - 3)}^{2} {(x - 2)}^{2}}$

Step 6.3.1.8

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

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Step 6.3.1.8.1

Apply the distributive property.

$\frac{2 x^{3} - 5 x^{2} - 13 x + 30 + (- x (x + 2) - 3 (x + 2)) (2 x - 5)}{{(x - 3)}^{2} {(x - 2)}^{2}}$

Step 6.3.1.8.2

Apply the distributive property.

$\frac{2 x^{3} - 5 x^{2} - 13 x + 30 + (- x \cdot x - x \cdot 2 - 3 (x + 2)) (2 x - 5)}{{(x - 3)}^{2} {(x - 2)}^{2}}$

Step 6.3.1.8.3

Apply the distributive property.

$\frac{2 x^{3} - 5 x^{2} - 13 x + 30 + (- x \cdot x - x \cdot 2 - 3 x - 3 \cdot 2) (2 x - 5)}{{(x - 3)}^{2} {(x - 2)}^{2}}$

Step 6.3.1.9

Simplify and combine like terms.

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Step 6.3.1.9.1

Simplify each term.

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Step 6.3.1.9.1.1

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

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Step 6.3.1.9.1.1.1

Move $x$ .

$\frac{2 x^{3} - 5 x^{2} - 13 x + 30 + (- (x \cdot x) - x \cdot 2 - 3 x - 3 \cdot 2) (2 x - 5)}{{(x - 3)}^{2} {(x - 2)}^{2}}$

Step 6.3.1.9.1.1.2

Multiply $x$ by $x$ .

$\frac{2 x^{3} - 5 x^{2} - 13 x + 30 + (- x^{2} - x \cdot 2 - 3 x - 3 \cdot 2) (2 x - 5)}{{(x - 3)}^{2} {(x - 2)}^{2}}$

Step 6.3.1.9.1.2

Multiply $2$ by $- 1$ .

$\frac{2 x^{3} - 5 x^{2} - 13 x + 30 + (- x^{2} - 2 x - 3 x - 3 \cdot 2) (2 x - 5)}{{(x - 3)}^{2} {(x - 2)}^{2}}$

Step 6.3.1.9.1.3

Multiply $- 3$ by $2$ .

$\frac{2 x^{3} - 5 x^{2} - 13 x + 30 + (- x^{2} - 2 x - 3 x - 6) (2 x - 5)}{{(x - 3)}^{2} {(x - 2)}^{2}}$

Step 6.3.1.9.2

Subtract $3 x$ from $- 2 x$ .

$\frac{2 x^{3} - 5 x^{2} - 13 x + 30 + (- x^{2} - 5 x - 6) (2 x - 5)}{{(x - 3)}^{2} {(x - 2)}^{2}}$

Step 6.3.1.10

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

$\frac{2 x^{3} - 5 x^{2} - 13 x + 30 - x^{2} (2 x) - x^{2} \cdot - 5 - 5 x (2 x) - 5 x \cdot - 5 - 6 (2 x) - 6 \cdot - 5}{{(x - 3)}^{2} {(x - 2)}^{2}}$

Step 6.3.1.11

Simplify each term.

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Step 6.3.1.11.1

Rewrite using the commutative property of multiplication.

$\frac{2 x^{3} - 5 x^{2} - 13 x + 30 - 1 \cdot 2 x^{2} x - x^{2} \cdot - 5 - 5 x (2 x) - 5 x \cdot - 5 - 6 (2 x) - 6 \cdot - 5}{{(x - 3)}^{2} {(x - 2)}^{2}}$

Step 6.3.1.11.2

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

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Step 6.3.1.11.2.1

Move $x$ .

$\frac{2 x^{3} - 5 x^{2} - 13 x + 30 - 1 \cdot 2 (x \cdot x^{2}) - x^{2} \cdot - 5 - 5 x (2 x) - 5 x \cdot - 5 - 6 (2 x) - 6 \cdot - 5}{{(x - 3)}^{2} {(x - 2)}^{2}}$

Step 6.3.1.11.2.2

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

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Step 6.3.1.11.2.2.1

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

$\frac{2 x^{3} - 5 x^{2} - 13 x + 30 - 1 \cdot 2 (x^{1} x^{2}) - x^{2} \cdot - 5 - 5 x (2 x) - 5 x \cdot - 5 - 6 (2 x) - 6 \cdot - 5}{{(x - 3)}^{2} {(x - 2)}^{2}}$

Step 6.3.1.11.2.2.2

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

$\frac{2 x^{3} - 5 x^{2} - 13 x + 30 - 1 \cdot 2 x^{1 + 2} - x^{2} \cdot - 5 - 5 x (2 x) - 5 x \cdot - 5 - 6 (2 x) - 6 \cdot - 5}{{(x - 3)}^{2} {(x - 2)}^{2}}$

Step 6.3.1.11.2.3

Add $1$ and $2$ .

$\frac{2 x^{3} - 5 x^{2} - 13 x + 30 - 1 \cdot 2 x^{3} - x^{2} \cdot - 5 - 5 x (2 x) - 5 x \cdot - 5 - 6 (2 x) - 6 \cdot - 5}{{(x - 3)}^{2} {(x - 2)}^{2}}$

Step 6.3.1.11.3

Multiply $- 1$ by $2$ .

$\frac{2 x^{3} - 5 x^{2} - 13 x + 30 - 2 x^{3} - x^{2} \cdot - 5 - 5 x (2 x) - 5 x \cdot - 5 - 6 (2 x) - 6 \cdot - 5}{{(x - 3)}^{2} {(x - 2)}^{2}}$

Step 6.3.1.11.4

Multiply $- 5$ by $- 1$ .

$\frac{2 x^{3} - 5 x^{2} - 13 x + 30 - 2 x^{3} + 5 x^{2} - 5 x (2 x) - 5 x \cdot - 5 - 6 (2 x) - 6 \cdot - 5}{{(x - 3)}^{2} {(x - 2)}^{2}}$

Step 6.3.1.11.5

Rewrite using the commutative property of multiplication.

$\frac{2 x^{3} - 5 x^{2} - 13 x + 30 - 2 x^{3} + 5 x^{2} - 5 \cdot 2 x \cdot x - 5 x \cdot - 5 - 6 (2 x) - 6 \cdot - 5}{{(x - 3)}^{2} {(x - 2)}^{2}}$

Step 6.3.1.11.6

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

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Step 6.3.1.11.6.1

Move $x$ .

$\frac{2 x^{3} - 5 x^{2} - 13 x + 30 - 2 x^{3} + 5 x^{2} - 5 \cdot 2 (x \cdot x) - 5 x \cdot - 5 - 6 (2 x) - 6 \cdot - 5}{{(x - 3)}^{2} {(x - 2)}^{2}}$

Step 6.3.1.11.6.2

Multiply $x$ by $x$ .

$\frac{2 x^{3} - 5 x^{2} - 13 x + 30 - 2 x^{3} + 5 x^{2} - 5 \cdot 2 x^{2} - 5 x \cdot - 5 - 6 (2 x) - 6 \cdot - 5}{{(x - 3)}^{2} {(x - 2)}^{2}}$

Step 6.3.1.11.7

Multiply $- 5$ by $2$ .

$\frac{2 x^{3} - 5 x^{2} - 13 x + 30 - 2 x^{3} + 5 x^{2} - 10 x^{2} - 5 x \cdot - 5 - 6 (2 x) - 6 \cdot - 5}{{(x - 3)}^{2} {(x - 2)}^{2}}$

Step 6.3.1.11.8

Multiply $- 5$ by $- 5$ .

$\frac{2 x^{3} - 5 x^{2} - 13 x + 30 - 2 x^{3} + 5 x^{2} - 10 x^{2} + 25 x - 6 (2 x) - 6 \cdot - 5}{{(x - 3)}^{2} {(x - 2)}^{2}}$

Step 6.3.1.11.9

Multiply $2$ by $- 6$ .

$\frac{2 x^{3} - 5 x^{2} - 13 x + 30 - 2 x^{3} + 5 x^{2} - 10 x^{2} + 25 x - 12 x - 6 \cdot - 5}{{(x - 3)}^{2} {(x - 2)}^{2}}$

Step 6.3.1.11.10

Multiply $- 6$ by $- 5$ .

$\frac{2 x^{3} - 5 x^{2} - 13 x + 30 - 2 x^{3} + 5 x^{2} - 10 x^{2} + 25 x - 12 x + 30}{{(x - 3)}^{2} {(x - 2)}^{2}}$

Step 6.3.1.12

Subtract $10 x^{2}$ from $5 x^{2}$ .

$\frac{2 x^{3} - 5 x^{2} - 13 x + 30 - 2 x^{3} - 5 x^{2} + 25 x - 12 x + 30}{{(x - 3)}^{2} {(x - 2)}^{2}}$

Step 6.3.1.13

Subtract $12 x$ from $25 x$ .

$\frac{2 x^{3} - 5 x^{2} - 13 x + 30 - 2 x^{3} - 5 x^{2} + 13 x + 30}{{(x - 3)}^{2} {(x - 2)}^{2}}$

Step 6.3.2

Combine the opposite terms in $2 x^{3} - 5 x^{2} - 13 x + 30 - 2 x^{3} - 5 x^{2} + 13 x + 30$ .

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Step 6.3.2.1

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

$\frac{- 5 x^{2} - 13 x + 30 + 0 - 5 x^{2} + 13 x + 30}{{(x - 3)}^{2} {(x - 2)}^{2}}$

Step 6.3.2.2

Add $- 5 x^{2} - 13 x + 30$ and $0$ .

$\frac{- 5 x^{2} - 13 x + 30 - 5 x^{2} + 13 x + 30}{{(x - 3)}^{2} {(x - 2)}^{2}}$

Step 6.3.2.3

Add $- 13 x$ and $13 x$ .

$\frac{- 5 x^{2} + 30 - 5 x^{2} + 0 + 30}{{(x - 3)}^{2} {(x - 2)}^{2}}$

Step 6.3.2.4

Add $- 5 x^{2} + 30 - 5 x^{2}$ and $0$ .

$\frac{- 5 x^{2} + 30 - 5 x^{2} + 30}{{(x - 3)}^{2} {(x - 2)}^{2}}$

Step 6.3.3

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

$\frac{- 10 x^{2} + 30 + 30}{{(x - 3)}^{2} {(x - 2)}^{2}}$

Step 6.3.4

Add $30$ and $30$ .

$\frac{- 10 x^{2} + 60}{{(x - 3)}^{2} {(x - 2)}^{2}}$

Step 6.4

Reorder terms.

$\frac{- 10 x^{2} + 60}{{(x - 2)}^{2} {(x - 3)}^{2}}$

Step 6.5

Factor $10$ out of $- 10 x^{2} + 60$ .

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Step 6.5.1

Factor $10$ out of $- 10 x^{2}$ .

$\frac{10 (- x^{2}) + 60}{{(x - 2)}^{2} {(x - 3)}^{2}}$

Step 6.5.2

Factor $10$ out of $60$ .

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

Step 6.5.3

Factor $10$ out of $10 (- x^{2}) + 10 (6)$ .

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

Step 6.6

Factor $- 1$ out of $- x^{2}$ .

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

Step 6.7

Rewrite $6$ as $- 1 (- 6)$ .

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

Step 6.8

Factor $- 1$ out of $- (x^{2}) - 1 (- 6)$ .

$\frac{10 (- (x^{2} - 6))}{{(x - 2)}^{2} {(x - 3)}^{2}}$

Step 6.9

Rewrite $- (x^{2} - 6)$ as $- 1 (x^{2} - 6)$ .

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

Step 6.10

Move the negative in front of the fraction.

$- \frac{10 (x^{2} - 6)}{{(x - 2)}^{2} {(x - 3)}^{2}}$