Calculus Examples

$\frac{- 2}{x^{2} - x - 6}$

Step 1

Write $\frac{- 2}{x^{2} - x - 6}$ as a function.

$f (x) = \frac{- 2}{x^{2} - x - 6}$

Step 2

Find the second derivative.

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Step 2.1

Find the first derivative.

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Step 2.1.1

Differentiate using the Constant Multiple Rule.

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Step 2.1.1.1

Move the negative in front of the fraction.

$\frac{d}{d x} [- \frac{2}{x^{2} - x - 6}]$

Step 2.1.1.2

Since $- 2$ is constant with respect to $x$ , the derivative of $- \frac{2}{x^{2} - x - 6}$ with respect to $x$ is $- 2 \frac{d}{d x} [\frac{1}{x^{2} - x - 6}]$ .

$- 2 \frac{d}{d x} [\frac{1}{x^{2} - x - 6}]$

Step 2.1.1.3

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

$- 2 \frac{d}{d x} [{(x^{2} - x - 6)}^{- 1}]$

Step 2.1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{2} - x - 6$ .

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Step 2.1.2.1

To apply the Chain Rule, set $u$ as $x^{2} - x - 6$ .

$- 2 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{2} - x - 6])$

Step 2.1.2.2

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

$- 2 (- u^{- 2} \frac{d}{d x} [x^{2} - x - 6])$

Step 2.1.2.3

Replace all occurrences of $u$ with $x^{2} - x - 6$ .

$- 2 (- {(x^{2} - x - 6)}^{- 2} \frac{d}{d x} [x^{2} - x - 6])$

Step 2.1.3

Differentiate.

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Step 2.1.3.1

Multiply $- 1$ by $- 2$ .

$2 ({(x^{2} - x - 6)}^{- 2} \frac{d}{d x} [x^{2} - x - 6])$

Step 2.1.3.2

By the Sum Rule, the derivative of $x^{2} - x - 6$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- x] + \frac{d}{d x} [- 6]$ .

$2 {(x^{2} - x - 6)}^{- 2} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- x] + \frac{d}{d x} [- 6])$

Step 2.1.3.3

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

$2 {(x^{2} - x - 6)}^{- 2} (2 x + \frac{d}{d x} [- x] + \frac{d}{d x} [- 6])$

Step 2.1.3.4

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$2 {(x^{2} - x - 6)}^{- 2} (2 x - \frac{d}{d x} [x] + \frac{d}{d x} [- 6])$

Step 2.1.3.5

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

$2 {(x^{2} - x - 6)}^{- 2} (2 x - 1 \cdot 1 + \frac{d}{d x} [- 6])$

Step 2.1.3.6

Multiply $- 1$ by $1$ .

$2 {(x^{2} - x - 6)}^{- 2} (2 x - 1 + \frac{d}{d x} [- 6])$

Step 2.1.3.7

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6$ with respect to $x$ is $0$ .

$2 {(x^{2} - x - 6)}^{- 2} (2 x - 1 + 0)$

Step 2.1.3.8

Add $2 x - 1$ and $0$ .

$2 {(x^{2} - x - 6)}^{- 2} (2 x - 1)$

Step 2.1.4

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2.1.5

Simplify.

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Step 2.1.5.1

Combine $2$ and $\frac{1}{{(x^{2} - x - 6)}^{2}}$ .

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

Step 2.1.5.2

Reorder the factors of $\frac{2}{{(x^{2} - x - 6)}^{2}} (2 x - 1)$ .

$f' (x) = (2 x - 1) \frac{2}{{(x^{2} - x - 6)}^{2}}$

Step 2.2

Find the second derivative.

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Step 2.2.1

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 $f (x) = 2 x - 1$ and $g (x) = \frac{2}{{(x^{2} - x - 6)}^{2}}$ .

$(2 x - 1) \frac{d}{d x} [\frac{2}{{(x^{2} - x - 6)}^{2}}] + \frac{2}{{(x^{2} - x - 6)}^{2}} \frac{d}{d x} [2 x - 1]$

Step 2.2.2

Differentiate using the Constant Multiple Rule.

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Step 2.2.2.1

Since $2$ is constant with respect to $x$ , the derivative of $\frac{2}{{(x^{2} - x - 6)}^{2}}$ with respect to $x$ is $2 \frac{d}{d x} [\frac{1}{{(x^{2} - x - 6)}^{2}}]$ .

$(2 x - 1) (2 \frac{d}{d x} [\frac{1}{{(x^{2} - x - 6)}^{2}}]) + \frac{2}{{(x^{2} - x - 6)}^{2}} \frac{d}{d x} [2 x - 1]$

Step 2.2.2.2

Simplify the expression.

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Step 2.2.2.2.1

Move $2$ to the left of $2 x - 1$ .

$2 \cdot (2 x - 1) \frac{d}{d x} [\frac{1}{{(x^{2} - x - 6)}^{2}}] + \frac{2}{{(x^{2} - x - 6)}^{2}} \frac{d}{d x} [2 x - 1]$

Step 2.2.2.2.2

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

$2 (2 x - 1) \frac{d}{d x} [{({(x^{2} - x - 6)}^{2})}^{- 1}] + \frac{2}{{(x^{2} - x - 6)}^{2}} \frac{d}{d x} [2 x - 1]$

Step 2.2.2.2.3

Multiply the exponents in ${({(x^{2} - x - 6)}^{2})}^{- 1}$ .

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Step 2.2.2.2.3.1

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

$2 (2 x - 1) \frac{d}{d x} [{(x^{2} - x - 6)}^{2 \cdot - 1}] + \frac{2}{{(x^{2} - x - 6)}^{2}} \frac{d}{d x} [2 x - 1]$

Step 2.2.2.2.3.2

Multiply $2$ by $- 1$ .

$2 (2 x - 1) \frac{d}{d x} [{(x^{2} - x - 6)}^{- 2}] + \frac{2}{{(x^{2} - x - 6)}^{2}} \frac{d}{d x} [2 x - 1]$

Step 2.2.3

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 $g (x) = x^{2} - x - 6$ .

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Step 2.2.3.1

To apply the Chain Rule, set $u$ as $x^{2} - x - 6$ .

$2 (2 x - 1) (\frac{d}{d u} [u^{- 2}] \frac{d}{d x} [x^{2} - x - 6]) + \frac{2}{{(x^{2} - x - 6)}^{2}} \frac{d}{d x} [2 x - 1]$

Step 2.2.3.2

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

$2 (2 x - 1) (- 2 u^{- 3} \frac{d}{d x} [x^{2} - x - 6]) + \frac{2}{{(x^{2} - x - 6)}^{2}} \frac{d}{d x} [2 x - 1]$

Step 2.2.3.3

Replace all occurrences of $u$ with $x^{2} - x - 6$ .

$2 (2 x - 1) (- 2 {(x^{2} - x - 6)}^{- 3} \frac{d}{d x} [x^{2} - x - 6]) + \frac{2}{{(x^{2} - x - 6)}^{2}} \frac{d}{d x} [2 x - 1]$

Step 2.2.4

Differentiate.

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Step 2.2.4.1

Multiply $- 2$ by $2$ .

$- 4 (2 x - 1) ({(x^{2} - x - 6)}^{- 3} \frac{d}{d x} [x^{2} - x - 6]) + \frac{2}{{(x^{2} - x - 6)}^{2}} \frac{d}{d x} [2 x - 1]$

Step 2.2.4.2

By the Sum Rule, the derivative of $x^{2} - x - 6$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- x] + \frac{d}{d x} [- 6]$ .

$- 4 (2 x - 1) ({(x^{2} - x - 6)}^{- 3} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- x] + \frac{d}{d x} [- 6])) + \frac{2}{{(x^{2} - x - 6)}^{2}} \frac{d}{d x} [2 x - 1]$

Step 2.2.4.3

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

$- 4 (2 x - 1) ({(x^{2} - x - 6)}^{- 3} (2 x + \frac{d}{d x} [- x] + \frac{d}{d x} [- 6])) + \frac{2}{{(x^{2} - x - 6)}^{2}} \frac{d}{d x} [2 x - 1]$

Step 2.2.4.4

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$- 4 (2 x - 1) ({(x^{2} - x - 6)}^{- 3} (2 x - \frac{d}{d x} [x] + \frac{d}{d x} [- 6])) + \frac{2}{{(x^{2} - x - 6)}^{2}} \frac{d}{d x} [2 x - 1]$

Step 2.2.4.5

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

$- 4 (2 x - 1) ({(x^{2} - x - 6)}^{- 3} (2 x - 1 \cdot 1 + \frac{d}{d x} [- 6])) + \frac{2}{{(x^{2} - x - 6)}^{2}} \frac{d}{d x} [2 x - 1]$

Step 2.2.4.6

Multiply $- 1$ by $1$ .

$- 4 (2 x - 1) ({(x^{2} - x - 6)}^{- 3} (2 x - 1 + \frac{d}{d x} [- 6])) + \frac{2}{{(x^{2} - x - 6)}^{2}} \frac{d}{d x} [2 x - 1]$

Step 2.2.4.7

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6$ with respect to $x$ is $0$ .

$- 4 (2 x - 1) ({(x^{2} - x - 6)}^{- 3} (2 x - 1 + 0)) + \frac{2}{{(x^{2} - x - 6)}^{2}} \frac{d}{d x} [2 x - 1]$

Step 2.2.4.8

Add $2 x - 1$ and $0$ .

$- 4 (2 x - 1) ({(x^{2} - x - 6)}^{- 3} (2 x - 1)) + \frac{2}{{(x^{2} - x - 6)}^{2}} \frac{d}{d x} [2 x - 1]$

Step 2.2.5

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

$- 4 ({(2 x - 1)}^{1} (2 x - 1)) {(x^{2} - x - 6)}^{- 3} + \frac{2}{{(x^{2} - x - 6)}^{2}} \frac{d}{d x} [2 x - 1]$

Step 2.2.6

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

$- 4 ({(2 x - 1)}^{1} {(2 x - 1)}^{1}) {(x^{2} - x - 6)}^{- 3} + \frac{2}{{(x^{2} - x - 6)}^{2}} \frac{d}{d x} [2 x - 1]$

Step 2.2.7

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

$- 4 {(2 x - 1)}^{1 + 1} {(x^{2} - x - 6)}^{- 3} + \frac{2}{{(x^{2} - x - 6)}^{2}} \frac{d}{d x} [2 x - 1]$

Step 2.2.8

Add $1$ and $1$ .

$- 4 {(2 x - 1)}^{2} {(x^{2} - x - 6)}^{- 3} + \frac{2}{{(x^{2} - x - 6)}^{2}} \frac{d}{d x} [2 x - 1]$

Step 2.2.9

By the Sum Rule, the derivative of $2 x - 1$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [- 1]$ .

$- 4 {(2 x - 1)}^{2} {(x^{2} - x - 6)}^{- 3} + \frac{2}{{(x^{2} - x - 6)}^{2}} (\frac{d}{d x} [2 x] + \frac{d}{d x} [- 1])$

Step 2.2.10

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$- 4 {(2 x - 1)}^{2} {(x^{2} - x - 6)}^{- 3} + \frac{2}{{(x^{2} - x - 6)}^{2}} (2 \frac{d}{d x} [x] + \frac{d}{d x} [- 1])$

Step 2.2.11

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

$- 4 {(2 x - 1)}^{2} {(x^{2} - x - 6)}^{- 3} + \frac{2}{{(x^{2} - x - 6)}^{2}} (2 \cdot 1 + \frac{d}{d x} [- 1])$

Step 2.2.12

Multiply $2$ by $1$ .

$- 4 {(2 x - 1)}^{2} {(x^{2} - x - 6)}^{- 3} + \frac{2}{{(x^{2} - x - 6)}^{2}} (2 + \frac{d}{d x} [- 1])$

Step 2.2.13

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

$- 4 {(2 x - 1)}^{2} {(x^{2} - x - 6)}^{- 3} + \frac{2}{{(x^{2} - x - 6)}^{2}} (2 + 0)$

Step 2.2.14

Combine fractions.

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Step 2.2.14.1

Add $2$ and $0$ .

$- 4 {(2 x - 1)}^{2} {(x^{2} - x - 6)}^{- 3} + \frac{2}{{(x^{2} - x - 6)}^{2}} \cdot 2$

Step 2.2.14.2

Combine $\frac{2}{{(x^{2} - x - 6)}^{2}}$ and $2$ .

$- 4 {(2 x - 1)}^{2} {(x^{2} - x - 6)}^{- 3} + \frac{2 \cdot 2}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.14.3

Multiply $2$ by $2$ .

$- 4 {(2 x - 1)}^{2} {(x^{2} - x - 6)}^{- 3} + \frac{4}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.15

To write $- 4 {(2 x - 1)}^{2} {(x^{2} - x - 6)}^{- 3}$ as a fraction with a common denominator, multiply by $\frac{{(x^{2} - x - 6)}^{2}}{{(x^{2} - x - 6)}^{2}}$ .

$- 4 {(2 x - 1)}^{2} {(x^{2} - x - 6)}^{- 3} \cdot \frac{{(x^{2} - x - 6)}^{2}}{{(x^{2} - x - 6)}^{2}} + \frac{4}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.16

Combine $- 4 {(2 x - 1)}^{2} {(x^{2} - x - 6)}^{- 3}$ and $\frac{{(x^{2} - x - 6)}^{2}}{{(x^{2} - x - 6)}^{2}}$ .

$\frac{- 4 {(2 x - 1)}^{2} {(x^{2} - x - 6)}^{- 3} {(x^{2} - x - 6)}^{2}}{{(x^{2} - x - 6)}^{2}} + \frac{4}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.17

Combine the numerators over the common denominator.

$\frac{- 4 {(2 x - 1)}^{2} {(x^{2} - x - 6)}^{- 3} {(x^{2} - x - 6)}^{2} + 4}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.18

Multiply ${(x^{2} - x - 6)}^{- 3}$ by ${(x^{2} - x - 6)}^{2}$ by adding the exponents.

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Step 2.2.18.1

Move ${(x^{2} - x - 6)}^{2}$ .

$\frac{- 4 {(2 x - 1)}^{2} ({(x^{2} - x - 6)}^{2} {(x^{2} - x - 6)}^{- 3}) + 4}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.18.2

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

$\frac{- 4 {(2 x - 1)}^{2} {(x^{2} - x - 6)}^{2 - 3} + 4}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.18.3

Subtract $3$ from $2$ .

$\frac{- 4 {(2 x - 1)}^{2} {(x^{2} - x - 6)}^{- 1} + 4}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19

Simplify.

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Step 2.2.19.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{- 4 {(2 x - 1)}^{2} \frac{1}{x^{2} - x - 6} + 4}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2

Simplify the numerator.

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Step 2.2.19.2.1

Simplify each term.

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Step 2.2.19.2.1.1

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

$\frac{- 4 ((2 x - 1) (2 x - 1)) \frac{1}{x^{2} - x - 6} + 4}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.1.2

Expand $(2 x - 1) (2 x - 1)$ using the FOIL Method.

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Step 2.2.19.2.1.2.1

Apply the distributive property.

$\frac{- 4 (2 x (2 x - 1) - 1 (2 x - 1)) \frac{1}{x^{2} - x - 6} + 4}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.1.2.2

Apply the distributive property.

$\frac{- 4 (2 x (2 x) + 2 x \cdot - 1 - 1 (2 x - 1)) \frac{1}{x^{2} - x - 6} + 4}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.1.2.3

Apply the distributive property.

$\frac{- 4 (2 x (2 x) + 2 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1) \frac{1}{x^{2} - x - 6} + 4}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.1.3

Simplify and combine like terms.

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Step 2.2.19.2.1.3.1

Simplify each term.

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Step 2.2.19.2.1.3.1.1

Rewrite using the commutative property of multiplication.

$\frac{- 4 (2 \cdot 2 x \cdot x + 2 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1) \frac{1}{x^{2} - x - 6} + 4}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.1.3.1.2

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

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Step 2.2.19.2.1.3.1.2.1

Move $x$ .

$\frac{- 4 (2 \cdot 2 (x \cdot x) + 2 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1) \frac{1}{x^{2} - x - 6} + 4}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.1.3.1.2.2

Multiply $x$ by $x$ .

$\frac{- 4 (2 \cdot 2 x^{2} + 2 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1) \frac{1}{x^{2} - x - 6} + 4}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.1.3.1.3

Multiply $2$ by $2$ .

$\frac{- 4 (4 x^{2} + 2 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1) \frac{1}{x^{2} - x - 6} + 4}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.1.3.1.4

Multiply $- 1$ by $2$ .

$\frac{- 4 (4 x^{2} - 2 x - 1 (2 x) - 1 \cdot - 1) \frac{1}{x^{2} - x - 6} + 4}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.1.3.1.5

Multiply $2$ by $- 1$ .

$\frac{- 4 (4 x^{2} - 2 x - 2 x - 1 \cdot - 1) \frac{1}{x^{2} - x - 6} + 4}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.1.3.1.6

Multiply $- 1$ by $- 1$ .

$\frac{- 4 (4 x^{2} - 2 x - 2 x + 1) \frac{1}{x^{2} - x - 6} + 4}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.1.3.2

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

$\frac{- 4 (4 x^{2} - 4 x + 1) \frac{1}{x^{2} - x - 6} + 4}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.1.4

Apply the distributive property.

$\frac{(- 4 (4 x^{2}) - 4 (- 4 x) - 4 \cdot 1) \frac{1}{x^{2} - x - 6} + 4}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.1.5

Simplify.

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Step 2.2.19.2.1.5.1

Multiply $4$ by $- 4$ .

$\frac{(- 16 x^{2} - 4 (- 4 x) - 4 \cdot 1) \frac{1}{x^{2} - x - 6} + 4}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.1.5.2

Multiply $- 4$ by $- 4$ .

$\frac{(- 16 x^{2} + 16 x - 4 \cdot 1) \frac{1}{x^{2} - x - 6} + 4}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.1.5.3

Multiply $- 4$ by $1$ .

$\frac{(- 16 x^{2} + 16 x - 4) \frac{1}{x^{2} - x - 6} + 4}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.1.6

Factor $x^{2} - x - 6$ using the AC method.

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Step 2.2.19.2.1.6.1

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 $- 6$ and whose sum is $- 1$ .

$- 3, 2$

Step 2.2.19.2.1.6.2

Write the factored form using these integers.

$\frac{(- 16 x^{2} + 16 x - 4) \frac{1}{(x - 3) (x + 2)} + 4}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.1.7

Multiply $- 16 x^{2} + 16 x - 4$ by $\frac{1}{(x - 3) (x + 2)}$ .

$\frac{\frac{- 16 x^{2} + 16 x - 4}{(x - 3) (x + 2)} + 4}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.1.8

Simplify the numerator.

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Step 2.2.19.2.1.8.1

Factor $4$ out of $- 16 x^{2} + 16 x - 4$ .

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Step 2.2.19.2.1.8.1.1

Factor $4$ out of $- 16 x^{2}$ .

$\frac{\frac{4 (- 4 x^{2}) + 16 x - 4}{(x - 3) (x + 2)} + 4}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.1.8.1.2

Factor $4$ out of $16 x$ .

$\frac{\frac{4 (- 4 x^{2}) + 4 (4 x) - 4}{(x - 3) (x + 2)} + 4}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.1.8.1.3

Factor $4$ out of $- 4$ .

$\frac{\frac{4 (- 4 x^{2}) + 4 (4 x) + 4 (- 1)}{(x - 3) (x + 2)} + 4}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.1.8.1.4

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

$\frac{\frac{4 (- 4 x^{2} + 4 x) + 4 (- 1)}{(x - 3) (x + 2)} + 4}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.1.8.1.5

Factor $4$ out of $4 (- 4 x^{2} + 4 x) + 4 (- 1)$ .

$\frac{\frac{4 (- 4 x^{2} + 4 x - 1)}{(x - 3) (x + 2)} + 4}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.1.8.2

Factor by grouping.

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Step 2.2.19.2.1.8.2.1

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 $a \cdot c = - 4 \cdot - 1 = 4$ and whose sum is $b = 4$ .

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Step 2.2.19.2.1.8.2.1.1

Factor $4$ out of $4 x$ .

$\frac{\frac{4 (- 4 x^{2} + 4 (x) - 1)}{(x - 3) (x + 2)} + 4}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.1.8.2.1.2

Rewrite $4$ as $2$ plus $2$

$\frac{\frac{4 (- 4 x^{2} + (2 + 2) x - 1)}{(x - 3) (x + 2)} + 4}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.1.8.2.1.3

Apply the distributive property.

$\frac{\frac{4 (- 4 x^{2} + 2 x + 2 x - 1)}{(x - 3) (x + 2)} + 4}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.1.8.2.2

Factor out the greatest common factor from each group.

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Step 2.2.19.2.1.8.2.2.1

Group the first two terms and the last two terms.

$\frac{\frac{4 ((- 4 x^{2} + 2 x) + 2 x - 1)}{(x - 3) (x + 2)} + 4}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.1.8.2.2.2

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

$\frac{\frac{4 (2 x (- 2 x + 1) - (- 2 x + 1))}{(x - 3) (x + 2)} + 4}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.1.8.2.3

Factor the polynomial by factoring out the greatest common factor, $- 2 x + 1$ .

$\frac{\frac{4 ((- 2 x + 1) (2 x - 1))}{(x - 3) (x + 2)} + 4}{{(x^{2} - x - 6)}^{2}}$

$\frac{\frac{4 (- 2 x + 1) (2 x - 1)}{(x - 3) (x + 2)} + 4}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.1.8.3

Combine exponents.

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Step 2.2.19.2.1.8.3.1

Factor $- 1$ out of $- 2 x$ .

$\frac{\frac{4 (- (2 x) + 1) (2 x - 1)}{(x - 3) (x + 2)} + 4}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.1.8.3.2

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

$\frac{\frac{4 (- (2 x) - 1 (- 1)) (2 x - 1)}{(x - 3) (x + 2)} + 4}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.1.8.3.3

Factor $- 1$ out of $- (2 x) - 1 (- 1)$ .

$\frac{\frac{4 (- (2 x - 1)) (2 x - 1)}{(x - 3) (x + 2)} + 4}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.1.8.3.4

Rewrite $- (2 x - 1)$ as $- 1 (2 x - 1)$ .

$\frac{\frac{4 (- 1 (2 x - 1)) (2 x - 1)}{(x - 3) (x + 2)} + 4}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.1.8.3.5

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

$\frac{\frac{4 \cdot - 1 ({(2 x - 1)}^{1} (2 x - 1))}{(x - 3) (x + 2)} + 4}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.1.8.3.6

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

$\frac{\frac{4 \cdot - 1 ({(2 x - 1)}^{1} {(2 x - 1)}^{1})}{(x - 3) (x + 2)} + 4}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.1.8.3.7

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

$\frac{\frac{4 \cdot - 1 {(2 x - 1)}^{1 + 1}}{(x - 3) (x + 2)} + 4}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.1.8.3.8

Add $1$ and $1$ .

$\frac{\frac{4 \cdot - 1 {(2 x - 1)}^{2}}{(x - 3) (x + 2)} + 4}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.1.8.3.9

Multiply $4$ by $- 1$ .

$\frac{\frac{- 4 {(2 x - 1)}^{2}}{(x - 3) (x + 2)} + 4}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.1.9

Move the negative in front of the fraction.

$\frac{- \frac{4 {(2 x - 1)}^{2}}{(x - 3) (x + 2)} + 4}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.2

To write $4$ as a fraction with a common denominator, multiply by $\frac{(x - 3) (x + 2)}{(x - 3) (x + 2)}$ .

$\frac{- \frac{4 {(2 x - 1)}^{2}}{(x - 3) (x + 2)} + 4 \cdot \frac{(x - 3) (x + 2)}{(x - 3) (x + 2)}}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.3

Combine $4$ and $\frac{(x - 3) (x + 2)}{(x - 3) (x + 2)}$ .

$\frac{- \frac{4 {(2 x - 1)}^{2}}{(x - 3) (x + 2)} + \frac{4 ((x - 3) (x + 2))}{(x - 3) (x + 2)}}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.4

Combine the numerators over the common denominator.

$\frac{\frac{- 4 {(2 x - 1)}^{2} + 4 ((x - 3) (x + 2))}{(x - 3) (x + 2)}}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.5

Simplify the numerator.

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Step 2.2.19.2.5.1

Factor $4$ out of $- 4 {(2 x - 1)}^{2} + 4 (x - 3) (x + 2)$ .

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Step 2.2.19.2.5.1.1

Factor $4$ out of $- 4 {(2 x - 1)}^{2}$ .

$\frac{\frac{4 (- {(2 x - 1)}^{2}) + 4 (x - 3) (x + 2)}{(x - 3) (x + 2)}}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.5.1.2

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

$\frac{\frac{4 (- {(2 x - 1)}^{2}) + 4 ((x - 3) (x + 2))}{(x - 3) (x + 2)}}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.5.1.3

Factor $4$ out of $4 (- {(2 x - 1)}^{2}) + 4 ((x - 3) (x + 2))$ .

$\frac{\frac{4 (- {(2 x - 1)}^{2} + (x - 3) (x + 2))}{(x - 3) (x + 2)}}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.5.2

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

$\frac{\frac{4 (- ((2 x - 1) (2 x - 1)) + (x - 3) (x + 2))}{(x - 3) (x + 2)}}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.5.3

Expand $(2 x - 1) (2 x - 1)$ using the FOIL Method.

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Step 2.2.19.2.5.3.1

Apply the distributive property.

$\frac{\frac{4 (- (2 x (2 x - 1) - 1 (2 x - 1)) + (x - 3) (x + 2))}{(x - 3) (x + 2)}}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.5.3.2

Apply the distributive property.

$\frac{\frac{4 (- (2 x (2 x) + 2 x \cdot - 1 - 1 (2 x - 1)) + (x - 3) (x + 2))}{(x - 3) (x + 2)}}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.5.3.3

Apply the distributive property.

$\frac{\frac{4 (- (2 x (2 x) + 2 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1) + (x - 3) (x + 2))}{(x - 3) (x + 2)}}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.5.4

Simplify and combine like terms.

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Step 2.2.19.2.5.4.1

Simplify each term.

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Step 2.2.19.2.5.4.1.1

Rewrite using the commutative property of multiplication.

$\frac{\frac{4 (- (2 \cdot 2 x \cdot x + 2 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1) + (x - 3) (x + 2))}{(x - 3) (x + 2)}}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.5.4.1.2

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

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Step 2.2.19.2.5.4.1.2.1

Move $x$ .

$\frac{\frac{4 (- (2 \cdot 2 (x \cdot x) + 2 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1) + (x - 3) (x + 2))}{(x - 3) (x + 2)}}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.5.4.1.2.2

Multiply $x$ by $x$ .

$\frac{\frac{4 (- (2 \cdot 2 x^{2} + 2 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1) + (x - 3) (x + 2))}{(x - 3) (x + 2)}}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.5.4.1.3

Multiply $2$ by $2$ .

$\frac{\frac{4 (- (4 x^{2} + 2 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1) + (x - 3) (x + 2))}{(x - 3) (x + 2)}}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.5.4.1.4

Multiply $- 1$ by $2$ .

$\frac{\frac{4 (- (4 x^{2} - 2 x - 1 (2 x) - 1 \cdot - 1) + (x - 3) (x + 2))}{(x - 3) (x + 2)}}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.5.4.1.5

Multiply $2$ by $- 1$ .

$\frac{\frac{4 (- (4 x^{2} - 2 x - 2 x - 1 \cdot - 1) + (x - 3) (x + 2))}{(x - 3) (x + 2)}}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.5.4.1.6

Multiply $- 1$ by $- 1$ .

$\frac{\frac{4 (- (4 x^{2} - 2 x - 2 x + 1) + (x - 3) (x + 2))}{(x - 3) (x + 2)}}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.5.4.2

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

$\frac{\frac{4 (- (4 x^{2} - 4 x + 1) + (x - 3) (x + 2))}{(x - 3) (x + 2)}}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.5.5

Apply the distributive property.

$\frac{\frac{4 (- (4 x^{2}) - (- 4 x) - 1 \cdot 1 + (x - 3) (x + 2))}{(x - 3) (x + 2)}}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.5.6

Simplify.

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Step 2.2.19.2.5.6.1

Multiply $4$ by $- 1$ .

$\frac{\frac{4 (- 4 x^{2} - (- 4 x) - 1 \cdot 1 + (x - 3) (x + 2))}{(x - 3) (x + 2)}}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.5.6.2

Multiply $- 4$ by $- 1$ .

$\frac{\frac{4 (- 4 x^{2} + 4 x - 1 \cdot 1 + (x - 3) (x + 2))}{(x - 3) (x + 2)}}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.5.6.3

Multiply $- 1$ by $1$ .

$\frac{\frac{4 (- 4 x^{2} + 4 x - 1 + (x - 3) (x + 2))}{(x - 3) (x + 2)}}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.5.7

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

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Step 2.2.19.2.5.7.1

Apply the distributive property.

$\frac{\frac{4 (- 4 x^{2} + 4 x - 1 + x (x + 2) - 3 (x + 2))}{(x - 3) (x + 2)}}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.5.7.2

Apply the distributive property.

$\frac{\frac{4 (- 4 x^{2} + 4 x - 1 + x \cdot x + x \cdot 2 - 3 (x + 2))}{(x - 3) (x + 2)}}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.5.7.3

Apply the distributive property.

$\frac{\frac{4 (- 4 x^{2} + 4 x - 1 + x \cdot x + x \cdot 2 - 3 x - 3 \cdot 2)}{(x - 3) (x + 2)}}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.5.8

Simplify and combine like terms.

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Step 2.2.19.2.5.8.1

Simplify each term.

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Step 2.2.19.2.5.8.1.1

Multiply $x$ by $x$ .

$\frac{\frac{4 (- 4 x^{2} + 4 x - 1 + x^{2} + x \cdot 2 - 3 x - 3 \cdot 2)}{(x - 3) (x + 2)}}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.5.8.1.2

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

$\frac{\frac{4 (- 4 x^{2} + 4 x - 1 + x^{2} + 2 \cdot x - 3 x - 3 \cdot 2)}{(x - 3) (x + 2)}}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.5.8.1.3

Multiply $- 3$ by $2$ .

$\frac{\frac{4 (- 4 x^{2} + 4 x - 1 + x^{2} + 2 x - 3 x - 6)}{(x - 3) (x + 2)}}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.5.8.2

Subtract $3 x$ from $2 x$ .

$\frac{\frac{4 (- 4 x^{2} + 4 x - 1 + x^{2} - x - 6)}{(x - 3) (x + 2)}}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.5.9

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

$\frac{\frac{4 (- 3 x^{2} + 4 x - 1 - x - 6)}{(x - 3) (x + 2)}}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.5.10

Subtract $x$ from $4 x$ .

$\frac{\frac{4 (- 3 x^{2} + 3 x - 1 - 6)}{(x - 3) (x + 2)}}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.5.11

Subtract $6$ from $- 1$ .

$\frac{\frac{4 (- 3 x^{2} + 3 x - 7)}{(x - 3) (x + 2)}}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.6

Factor $- 1$ out of $- 3 x^{2}$ .

$\frac{\frac{4 (- (3 x^{2}) + 3 x - 7)}{(x - 3) (x + 2)}}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.7

Factor $- 1$ out of $3 x$ .

$\frac{\frac{4 (- (3 x^{2}) - (- 3 x) - 7)}{(x - 3) (x + 2)}}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.8

Factor $- 1$ out of $- (3 x^{2}) - (- 3 x)$ .

$\frac{\frac{4 (- (3 x^{2} - 3 x) - 7)}{(x - 3) (x + 2)}}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.9

Rewrite $- 7$ as $- 1 (7)$ .

$\frac{\frac{4 (- (3 x^{2} - 3 x) - 1 (7))}{(x - 3) (x + 2)}}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.10

Factor $- 1$ out of $- (3 x^{2} - 3 x) - 1 (7)$ .

$\frac{\frac{4 (- (3 x^{2} - 3 x + 7))}{(x - 3) (x + 2)}}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.11

Rewrite $- (3 x^{2} - 3 x + 7)$ as $- 1 (3 x^{2} - 3 x + 7)$ .

$\frac{\frac{4 (- 1 (3 x^{2} - 3 x + 7))}{(x - 3) (x + 2)}}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.2.12

Move the negative in front of the fraction.

$\frac{- \frac{4 (3 x^{2} - 3 x + 7)}{(x - 3) (x + 2)}}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.3

Combine terms.

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Step 2.2.19.3.1

Rewrite $\frac{- \frac{4 (3 x^{2} - 3 x + 7)}{(x - 3) (x + 2)}}{{(x^{2} - x - 6)}^{2}}$ as a product.

$- \frac{4 (3 x^{2} - 3 x + 7)}{(x - 3) (x + 2)} \cdot \frac{1}{{(x^{2} - x - 6)}^{2}}$

Step 2.2.19.3.2

Multiply $\frac{1}{{(x^{2} - x - 6)}^{2}}$ by $\frac{4 (3 x^{2} - 3 x + 7)}{(x - 3) (x + 2)}$ .

$- \frac{4 (3 x^{2} - 3 x + 7)}{{(x^{2} - x - 6)}^{2} (x - 3) (x + 2)}$

Step 2.2.19.4

Simplify the denominator.

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Step 2.2.19.4.1

Factor $x^{2} - x - 6$ using the AC method.

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Step 2.2.19.4.1.1

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 $- 6$ and whose sum is $- 1$ .

$- 3, 2$

Step 2.2.19.4.1.2

Write the factored form using these integers.

$- \frac{4 (3 x^{2} - 3 x + 7)}{{((x - 3) (x + 2))}^{2} (x - 3) (x + 2)}$

Step 2.2.19.4.2

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

$- \frac{4 (3 x^{2} - 3 x + 7)}{{(x - 3)}^{2} {(x + 2)}^{2} (x - 3) (x + 2)}$

Step 2.2.19.4.3

Combine exponents.

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Step 2.2.19.4.3.1

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

$- \frac{4 (3 x^{2} - 3 x + 7)}{{(x - 3)}^{1} {(x - 3)}^{2} {(x + 2)}^{2} (x + 2)}$

Step 2.2.19.4.3.2

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

$- \frac{4 (3 x^{2} - 3 x + 7)}{{(x - 3)}^{1 + 2} {(x + 2)}^{2} (x + 2)}$

Step 2.2.19.4.3.3

Add $1$ and $2$ .

$- \frac{4 (3 x^{2} - 3 x + 7)}{{(x - 3)}^{3} {(x + 2)}^{2} (x + 2)}$

Step 2.2.19.4.3.4

Raise $x + 2$ to the power of $1$ .

$- \frac{4 (3 x^{2} - 3 x + 7)}{{(x - 3)}^{3} ({(x + 2)}^{1} {(x + 2)}^{2})}$

Step 2.2.19.4.3.5

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

$- \frac{4 (3 x^{2} - 3 x + 7)}{{(x - 3)}^{3} {(x + 2)}^{1 + 2}}$

Step 2.2.19.4.3.6

Add $1$ and $2$ .

$f'' (x) = - \frac{4 (3 x^{2} - 3 x + 7)}{{(x - 3)}^{3} {(x + 2)}^{3}}$

Step 2.3

The second derivative of $f (x)$ with respect to $x$ is $- \frac{4 (3 x^{2} - 3 x + 7)}{{(x - 3)}^{3} {(x + 2)}^{3}}$ .

$- \frac{4 (3 x^{2} - 3 x + 7)}{{(x - 3)}^{3} {(x + 2)}^{3}}$

Step 3

Set the second derivative equal to $0$ then solve the equation $- \frac{4 (3 x^{2} - 3 x + 7)}{{(x - 3)}^{3} {(x + 2)}^{3}} = 0$ .

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Step 3.1

Set the second derivative equal to $0$ .

$- \frac{4 (3 x^{2} - 3 x + 7)}{{(x - 3)}^{3} {(x + 2)}^{3}} = 0$

Step 3.2

Set the numerator equal to zero.

$4 (3 x^{2} - 3 x + 7) = 0$

Step 3.3

Solve the equation for $x$ .

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Step 3.3.1

Divide each term in $4 (3 x^{2} - 3 x + 7) = 0$ by $4$ and simplify.

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Step 3.3.1.1

Divide each term in $4 (3 x^{2} - 3 x + 7) = 0$ by $4$ .

$\frac{4 (3 x^{2} - 3 x + 7)}{4} = \frac{0}{4}$

Step 3.3.1.2

Simplify the left side.

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Step 3.3.1.2.1

Cancel the common factor of $4$ .

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Step 3.3.1.2.1.1

Cancel the common factor.

$\frac{4 (3 x^{2} - 3 x + 7)}{4} = \frac{0}{4}$

Step 3.3.1.2.1.2

Divide $3 x^{2} - 3 x + 7$ by $1$ .

$3 x^{2} - 3 x + 7 = \frac{0}{4}$

Step 3.3.1.3

Simplify the right side.

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Step 3.3.1.3.1

Divide $0$ by $4$ .

$3 x^{2} - 3 x + 7 = 0$

Step 3.3.2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3.3.3

Substitute the values $a = 3$ , $b = - 3$ , and $c = 7$ into the quadratic formula and solve for $x$ .

$\frac{3 \pm \sqrt{{(- 3)}^{2} - 4 \cdot (3 \cdot 7)}}{2 \cdot 3}$

Step 3.3.4

Simplify.

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Step 3.3.4.1

Simplify the numerator.

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Step 3.3.4.1.1

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

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 3 \cdot 7}}{2 \cdot 3}$

Step 3.3.4.1.2

Multiply $- 4 \cdot 3 \cdot 7$ .

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Step 3.3.4.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{3 \pm \sqrt{9 - 12 \cdot 7}}{2 \cdot 3}$

Step 3.3.4.1.2.2

Multiply $- 12$ by $7$ .

$x = \frac{3 \pm \sqrt{9 - 84}}{2 \cdot 3}$

Step 3.3.4.1.3

Subtract $84$ from $9$ .

$x = \frac{3 \pm \sqrt{- 75}}{2 \cdot 3}$

Step 3.3.4.1.4

Rewrite $- 75$ as $- 1 (75)$ .

$x = \frac{3 \pm \sqrt{- 1 \cdot 75}}{2 \cdot 3}$

Step 3.3.4.1.5

Rewrite $\sqrt{- 1 (75)}$ as $\sqrt{- 1} \cdot \sqrt{75}$ .

$x = \frac{3 \pm \sqrt{- 1} \cdot \sqrt{75}}{2 \cdot 3}$

Step 3.3.4.1.6

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

$x = \frac{3 \pm i \cdot \sqrt{75}}{2 \cdot 3}$

Step 3.3.4.1.7

Rewrite $75$ as $5^{2} \cdot 3$ .

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Step 3.3.4.1.7.1

Factor $25$ out of $75$ .

$x = \frac{3 \pm i \cdot \sqrt{25 (3)}}{2 \cdot 3}$

Step 3.3.4.1.7.2

Rewrite $25$ as $5^{2}$ .

$x = \frac{3 \pm i \cdot \sqrt{5^{2} \cdot 3}}{2 \cdot 3}$

Step 3.3.4.1.8

Pull terms out from under the radical.

$x = \frac{3 \pm i \cdot (5 \sqrt{3})}{2 \cdot 3}$

Step 3.3.4.1.9

Move $5$ to the left of $i$ .

$x = \frac{3 \pm 5 i \sqrt{3}}{2 \cdot 3}$

Step 3.3.4.2

Multiply $2$ by $3$ .

$x = \frac{3 \pm 5 i \sqrt{3}}{6}$

Step 3.3.5

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 3.3.5.1

Simplify the numerator.

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Step 3.3.5.1.1

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

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 3 \cdot 7}}{2 \cdot 3}$

Step 3.3.5.1.2

Multiply $- 4 \cdot 3 \cdot 7$ .

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Step 3.3.5.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{3 \pm \sqrt{9 - 12 \cdot 7}}{2 \cdot 3}$

Step 3.3.5.1.2.2

Multiply $- 12$ by $7$ .

$x = \frac{3 \pm \sqrt{9 - 84}}{2 \cdot 3}$

Step 3.3.5.1.3

Subtract $84$ from $9$ .

$x = \frac{3 \pm \sqrt{- 75}}{2 \cdot 3}$

Step 3.3.5.1.4

Rewrite $- 75$ as $- 1 (75)$ .

$x = \frac{3 \pm \sqrt{- 1 \cdot 75}}{2 \cdot 3}$

Step 3.3.5.1.5

Rewrite $\sqrt{- 1 (75)}$ as $\sqrt{- 1} \cdot \sqrt{75}$ .

$x = \frac{3 \pm \sqrt{- 1} \cdot \sqrt{75}}{2 \cdot 3}$

Step 3.3.5.1.6

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

$x = \frac{3 \pm i \cdot \sqrt{75}}{2 \cdot 3}$

Step 3.3.5.1.7

Rewrite $75$ as $5^{2} \cdot 3$ .

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Step 3.3.5.1.7.1

Factor $25$ out of $75$ .

$x = \frac{3 \pm i \cdot \sqrt{25 (3)}}{2 \cdot 3}$

Step 3.3.5.1.7.2

Rewrite $25$ as $5^{2}$ .

$x = \frac{3 \pm i \cdot \sqrt{5^{2} \cdot 3}}{2 \cdot 3}$

Step 3.3.5.1.8

Pull terms out from under the radical.

$x = \frac{3 \pm i \cdot (5 \sqrt{3})}{2 \cdot 3}$

Step 3.3.5.1.9

Move $5$ to the left of $i$ .

$x = \frac{3 \pm 5 i \sqrt{3}}{2 \cdot 3}$

Step 3.3.5.2

Multiply $2$ by $3$ .

$x = \frac{3 \pm 5 i \sqrt{3}}{6}$

Step 3.3.5.3

Change the $\pm$ to $+$ .

$x = \frac{3 + 5 i \sqrt{3}}{6}$

Step 3.3.6

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 3.3.6.1

Simplify the numerator.

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Step 3.3.6.1.1

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

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 3 \cdot 7}}{2 \cdot 3}$

Step 3.3.6.1.2

Multiply $- 4 \cdot 3 \cdot 7$ .

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Step 3.3.6.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{3 \pm \sqrt{9 - 12 \cdot 7}}{2 \cdot 3}$

Step 3.3.6.1.2.2

Multiply $- 12$ by $7$ .

$x = \frac{3 \pm \sqrt{9 - 84}}{2 \cdot 3}$

Step 3.3.6.1.3

Subtract $84$ from $9$ .

$x = \frac{3 \pm \sqrt{- 75}}{2 \cdot 3}$

Step 3.3.6.1.4

Rewrite $- 75$ as $- 1 (75)$ .

$x = \frac{3 \pm \sqrt{- 1 \cdot 75}}{2 \cdot 3}$

Step 3.3.6.1.5

Rewrite $\sqrt{- 1 (75)}$ as $\sqrt{- 1} \cdot \sqrt{75}$ .

$x = \frac{3 \pm \sqrt{- 1} \cdot \sqrt{75}}{2 \cdot 3}$

Step 3.3.6.1.6

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

$x = \frac{3 \pm i \cdot \sqrt{75}}{2 \cdot 3}$

Step 3.3.6.1.7

Rewrite $75$ as $5^{2} \cdot 3$ .

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Step 3.3.6.1.7.1

Factor $25$ out of $75$ .

$x = \frac{3 \pm i \cdot \sqrt{25 (3)}}{2 \cdot 3}$

Step 3.3.6.1.7.2

Rewrite $25$ as $5^{2}$ .

$x = \frac{3 \pm i \cdot \sqrt{5^{2} \cdot 3}}{2 \cdot 3}$

Step 3.3.6.1.8

Pull terms out from under the radical.

$x = \frac{3 \pm i \cdot (5 \sqrt{3})}{2 \cdot 3}$

Step 3.3.6.1.9

Move $5$ to the left of $i$ .

$x = \frac{3 \pm 5 i \sqrt{3}}{2 \cdot 3}$

Step 3.3.6.2

Multiply $2$ by $3$ .

$x = \frac{3 \pm 5 i \sqrt{3}}{6}$

Step 3.3.6.3

Change the $\pm$ to $-$ .

$x = \frac{3 - 5 i \sqrt{3}}{6}$

Step 3.3.7

The final answer is the combination of both solutions.

$x = \frac{3 + 5 i \sqrt{3}}{6}, \frac{3 - 5 i \sqrt{3}}{6}$

Step 4

No values found that can make the second derivative equal to $0$ .

No Inflection Points