Calculus Examples

$y = \frac{x^{2} - 1}{x^{2} + x + 1}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $1$ as $1^{2}$ .

$y = \frac{x^{2} - 1^{2}}{x^{2} + x + 1}$

Step 1.2

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 $b = 1$ .

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

Step 2

Set $y$ as a function of $x$ .

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

Step 3

Find the derivative.

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Step 3.1

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

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

Step 3.2

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 $f (x) = x + 1$ and $g (x) = x - 1$ .

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

Step 3.3

Differentiate.

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

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

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

Step 3.3.4

Simplify the expression.

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Step 3.3.4.1

Add $1$ and $0$ .

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

Step 3.3.4.2

Multiply $x + 1$ by $1$ .

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

Step 3.3.5

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

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

Step 3.3.6

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

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

Step 3.3.7

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

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

Step 3.3.8

Simplify by adding terms.

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Step 3.3.8.1

Add $1$ and $0$ .

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

Step 3.3.8.2

Multiply $x - 1$ by $1$ .

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

Step 3.3.8.3

Add $x$ and $x$ .

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

Step 3.3.8.4

Simplify the expression.

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Step 3.3.8.4.1

Subtract $1$ from $1$ .

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

Step 3.3.8.4.2

Add $2 x$ and $0$ .

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

Step 3.3.8.4.3

Move $2$ to the left of $x^{2} + x + 1$ .

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

Step 3.3.9

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

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

Step 3.3.10

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

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

Step 3.3.11

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

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

Step 3.3.12

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

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

Step 3.3.13

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

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

Step 3.4

Simplify.

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Step 3.4.1

Apply the distributive property.

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

Step 3.4.2

Apply the distributive property.

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

Step 3.4.3

Apply the distributive property.

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

Step 3.4.4

Simplify the numerator.

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Step 3.4.4.1

Simplify each term.

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Step 3.4.4.1.1

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

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Step 3.4.4.1.1.1

Move $x$ .

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

Step 3.4.4.1.1.2

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

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Step 3.4.4.1.1.2.1

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

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

Step 3.4.4.1.1.2.2

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

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

Step 3.4.4.1.1.3

Add $1$ and $2$ .

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

Step 3.4.4.1.2

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

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Step 3.4.4.1.2.1

Move $x$ .

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

Step 3.4.4.1.2.2

Multiply $x$ by $x$ .

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

Step 3.4.4.1.3

Multiply $2$ by $1$ .

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

Step 3.4.4.1.4

Multiply $- 1$ by $1$ .

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

Step 3.4.4.1.5

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

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Step 3.4.4.1.5.1

Apply the distributive property.

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

Step 3.4.4.1.5.2

Apply the distributive property.

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

Step 3.4.4.1.5.3

Apply the distributive property.

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

Step 3.4.4.1.6

Simplify and combine like terms.

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Step 3.4.4.1.6.1

Simplify each term.

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Step 3.4.4.1.6.1.1

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

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Step 3.4.4.1.6.1.1.1

Move $x$ .

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

Step 3.4.4.1.6.1.1.2

Multiply $x$ by $x$ .

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

Step 3.4.4.1.6.1.2

Multiply $- x \cdot - 1$ .

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Step 3.4.4.1.6.1.2.1

Multiply $- 1$ by $- 1$ .

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

Step 3.4.4.1.6.1.2.2

Multiply $x$ by $1$ .

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

Step 3.4.4.1.6.1.3

Rewrite $- 1 x$ as $- x$ .

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

Step 3.4.4.1.6.1.4

Multiply $- 1$ by $- 1$ .

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

Step 3.4.4.1.6.2

Subtract $x$ from $x$ .

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

Step 3.4.4.1.6.3

Add $- x^{2}$ and $0$ .

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

Step 3.4.4.1.7

Expand $(- x^{2} + 1) (2 x + 1)$ using the FOIL Method.

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Step 3.4.4.1.7.1

Apply the distributive property.

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

Step 3.4.4.1.7.2

Apply the distributive property.

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

Step 3.4.4.1.7.3

Apply the distributive property.

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

Step 3.4.4.1.8

Simplify each term.

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Step 3.4.4.1.8.1

Rewrite using the commutative property of multiplication.

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

Step 3.4.4.1.8.2

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

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Step 3.4.4.1.8.2.1

Move $x$ .

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

Step 3.4.4.1.8.2.2

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

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Step 3.4.4.1.8.2.2.1

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

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

Step 3.4.4.1.8.2.2.2

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

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

Step 3.4.4.1.8.2.3

Add $1$ and $2$ .

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

Step 3.4.4.1.8.3

Multiply $- 1$ by $2$ .

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

Step 3.4.4.1.8.4

Multiply $- 1$ by $1$ .

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

Step 3.4.4.1.8.5

Multiply $2 x$ by $1$ .

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

Step 3.4.4.1.8.6

Multiply $1$ by $1$ .

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

Step 3.4.4.2

Combine the opposite terms in $2 x^{3} + 2 x^{2} + 2 x - 2 x^{3} - x^{2} + 2 x + 1$ .

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Step 3.4.4.2.1

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

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

Step 3.4.4.2.2

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

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

Step 3.4.4.3

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

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

Step 3.4.4.4

Add $2 x$ and $2 x$ .

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

Step 4

Set the derivative equal to $0$ then solve the equation $\frac{x^{2} + 4 x + 1}{{(x^{2} + x + 1)}^{2}} = 0$ .

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Step 4.1

Set the numerator equal to zero.

$x^{2} + 4 x + 1 = 0$

Step 4.2

Solve the equation for $x$ .

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Step 4.2.1

Use the quadratic formula to find the solutions.

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

Step 4.2.2

Substitute the values $a = 1$ , $b = 4$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{- 4 \pm \sqrt{4^{2} - 4 \cdot (1 \cdot 1)}}{2 \cdot 1}$

Step 4.2.3

Simplify.

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Step 4.2.3.1

Simplify the numerator.

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Step 4.2.3.1.1

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

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 4.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 1$ .

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Step 4.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1}}{2 \cdot 1}$

Step 4.2.3.1.2.2

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4}}{2 \cdot 1}$

Step 4.2.3.1.3

Subtract $4$ from $16$ .

$x = \frac{- 4 \pm \sqrt{12}}{2 \cdot 1}$

Step 4.2.3.1.4

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

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Step 4.2.3.1.4.1

Factor $4$ out of $12$ .

$x = \frac{- 4 \pm \sqrt{4 (3)}}{2 \cdot 1}$

Step 4.2.3.1.4.2

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

$x = \frac{- 4 \pm \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

Step 4.2.3.1.5

Pull terms out from under the radical.

$x = \frac{- 4 \pm 2 \sqrt{3}}{2 \cdot 1}$

Step 4.2.3.2

Multiply $2$ by $1$ .

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

Step 4.2.3.3

Simplify $\frac{- 4 \pm 2 \sqrt{3}}{2}$ .

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

Step 4.2.4

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

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Step 4.2.4.1

Simplify the numerator.

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Step 4.2.4.1.1

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

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 4.2.4.1.2

Multiply $- 4 \cdot 1 \cdot 1$ .

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Step 4.2.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1}}{2 \cdot 1}$

Step 4.2.4.1.2.2

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4}}{2 \cdot 1}$

Step 4.2.4.1.3

Subtract $4$ from $16$ .

$x = \frac{- 4 \pm \sqrt{12}}{2 \cdot 1}$

Step 4.2.4.1.4

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

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Step 4.2.4.1.4.1

Factor $4$ out of $12$ .

$x = \frac{- 4 \pm \sqrt{4 (3)}}{2 \cdot 1}$

Step 4.2.4.1.4.2

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

$x = \frac{- 4 \pm \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

Step 4.2.4.1.5

Pull terms out from under the radical.

$x = \frac{- 4 \pm 2 \sqrt{3}}{2 \cdot 1}$

Step 4.2.4.2

Multiply $2$ by $1$ .

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

Step 4.2.4.3

Simplify $\frac{- 4 \pm 2 \sqrt{3}}{2}$ .

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

Step 4.2.4.4

Change the $\pm$ to $+$ .

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

Step 4.2.5

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

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Step 4.2.5.1

Simplify the numerator.

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Step 4.2.5.1.1

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

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 4.2.5.1.2

Multiply $- 4 \cdot 1 \cdot 1$ .

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Step 4.2.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1}}{2 \cdot 1}$

Step 4.2.5.1.2.2

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4}}{2 \cdot 1}$

Step 4.2.5.1.3

Subtract $4$ from $16$ .

$x = \frac{- 4 \pm \sqrt{12}}{2 \cdot 1}$

Step 4.2.5.1.4

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

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Step 4.2.5.1.4.1

Factor $4$ out of $12$ .

$x = \frac{- 4 \pm \sqrt{4 (3)}}{2 \cdot 1}$

Step 4.2.5.1.4.2

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

$x = \frac{- 4 \pm \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

Step 4.2.5.1.5

Pull terms out from under the radical.

$x = \frac{- 4 \pm 2 \sqrt{3}}{2 \cdot 1}$

Step 4.2.5.2

Multiply $2$ by $1$ .

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

Step 4.2.5.3

Simplify $\frac{- 4 \pm 2 \sqrt{3}}{2}$ .

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

Step 4.2.5.4

Change the $\pm$ to $-$ .

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

Step 4.2.6

The final answer is the combination of both solutions.

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

Step 5

Solve the original function $f (x) = \frac{(x + 1) (x - 1)}{x^{2} + x + 1}$ at $x = - 2 + \sqrt{3}$ .

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Step 5.1

Replace the variable $x$ with $- 2 + \sqrt{3}$ in the expression.

$f (- 2 + \sqrt{3}) = \frac{((- 2 + \sqrt{3}) + 1) ((- 2 + \sqrt{3}) - 1)}{{(- 2 + \sqrt{3})}^{2} - 2 + \sqrt{3} + 1}$

Step 5.2

Simplify the result.

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Step 5.2.1

Remove parentheses.

$f (- 2 + \sqrt{3}) = \frac{((- 2 + \sqrt{3}) + 1) ((- 2 + \sqrt{3}) - 1)}{{(- 2 + \sqrt{3})}^{2} - 2 + \sqrt{3} + 1}$

Step 5.2.2

Simplify the numerator.

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Step 5.2.2.1

Add $- 2$ and $1$ .

$f (- 2 + \sqrt{3}) = \frac{(- 1 + \sqrt{3}) (- 2 + \sqrt{3} - 1)}{{(- 2 + \sqrt{3})}^{2} - 2 + \sqrt{3} + 1}$

Step 5.2.2.2

Subtract $1$ from $- 2$ .

$f (- 2 + \sqrt{3}) = \frac{(- 1 + \sqrt{3}) (- 3 + \sqrt{3})}{{(- 2 + \sqrt{3})}^{2} - 2 + \sqrt{3} + 1}$

Step 5.2.3

Simplify the denominator.

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Step 5.2.3.1

Rewrite ${(- 2 + \sqrt{3})}^{2}$ as $(- 2 + \sqrt{3}) (- 2 + \sqrt{3})$ .

$f (- 2 + \sqrt{3}) = \frac{(- 1 + \sqrt{3}) (- 3 + \sqrt{3})}{(- 2 + \sqrt{3}) (- 2 + \sqrt{3}) - 2 + \sqrt{3} + 1}$

Step 5.2.3.2

Expand $(- 2 + \sqrt{3}) (- 2 + \sqrt{3})$ using the FOIL Method.

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Step 5.2.3.2.1

Apply the distributive property.

$f (- 2 + \sqrt{3}) = \frac{(- 1 + \sqrt{3}) (- 3 + \sqrt{3})}{- 2 (- 2 + \sqrt{3}) + \sqrt{3} (- 2 + \sqrt{3}) - 2 + \sqrt{3} + 1}$

Step 5.2.3.2.2

Apply the distributive property.

$f (- 2 + \sqrt{3}) = \frac{(- 1 + \sqrt{3}) (- 3 + \sqrt{3})}{- 2 \cdot - 2 - 2 \sqrt{3} + \sqrt{3} (- 2 + \sqrt{3}) - 2 + \sqrt{3} + 1}$

Step 5.2.3.2.3

Apply the distributive property.

$f (- 2 + \sqrt{3}) = \frac{(- 1 + \sqrt{3}) (- 3 + \sqrt{3})}{- 2 \cdot - 2 - 2 \sqrt{3} + \sqrt{3} \cdot - 2 + \sqrt{3} \sqrt{3} - 2 + \sqrt{3} + 1}$

Step 5.2.3.3

Simplify and combine like terms.

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Step 5.2.3.3.1

Simplify each term.

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Step 5.2.3.3.1.1

Multiply $- 2$ by $- 2$ .

$f (- 2 + \sqrt{3}) = \frac{(- 1 + \sqrt{3}) (- 3 + \sqrt{3})}{4 - 2 \sqrt{3} + \sqrt{3} \cdot - 2 + \sqrt{3} \sqrt{3} - 2 + \sqrt{3} + 1}$

Step 5.2.3.3.1.2

Move $- 2$ to the left of $\sqrt{3}$ .

$f (- 2 + \sqrt{3}) = \frac{(- 1 + \sqrt{3}) (- 3 + \sqrt{3})}{4 - 2 \sqrt{3} - 2 \cdot \sqrt{3} + \sqrt{3} \sqrt{3} - 2 + \sqrt{3} + 1}$

Step 5.2.3.3.1.3

Combine using the product rule for radicals.

$f (- 2 + \sqrt{3}) = \frac{(- 1 + \sqrt{3}) (- 3 + \sqrt{3})}{4 - 2 \sqrt{3} - 2 \sqrt{3} + \sqrt{3 \cdot 3} - 2 + \sqrt{3} + 1}$

Step 5.2.3.3.1.4

Multiply $3$ by $3$ .

$f (- 2 + \sqrt{3}) = \frac{(- 1 + \sqrt{3}) (- 3 + \sqrt{3})}{4 - 2 \sqrt{3} - 2 \sqrt{3} + \sqrt{9} - 2 + \sqrt{3} + 1}$

Step 5.2.3.3.1.5

Rewrite $9$ as $3^{2}$ .

$f (- 2 + \sqrt{3}) = \frac{(- 1 + \sqrt{3}) (- 3 + \sqrt{3})}{4 - 2 \sqrt{3} - 2 \sqrt{3} + \sqrt{3^{2}} - 2 + \sqrt{3} + 1}$

Step 5.2.3.3.1.6

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

$f (- 2 + \sqrt{3}) = \frac{(- 1 + \sqrt{3}) (- 3 + \sqrt{3})}{4 - 2 \sqrt{3} - 2 \sqrt{3} + 3 - 2 + \sqrt{3} + 1}$

Step 5.2.3.3.2

Add $4$ and $3$ .

$f (- 2 + \sqrt{3}) = \frac{(- 1 + \sqrt{3}) (- 3 + \sqrt{3})}{7 - 2 \sqrt{3} - 2 \sqrt{3} - 2 + \sqrt{3} + 1}$

Step 5.2.3.3.3

Subtract $2 \sqrt{3}$ from $- 2 \sqrt{3}$ .

$f (- 2 + \sqrt{3}) = \frac{(- 1 + \sqrt{3}) (- 3 + \sqrt{3})}{7 - 4 \sqrt{3} - 2 + \sqrt{3} + 1}$

Step 5.2.3.4

Subtract $2$ from $7$ .

$f (- 2 + \sqrt{3}) = \frac{(- 1 + \sqrt{3}) (- 3 + \sqrt{3})}{5 - 4 \sqrt{3} + \sqrt{3} + 1}$

Step 5.2.3.5

Add $5$ and $1$ .

$f (- 2 + \sqrt{3}) = \frac{(- 1 + \sqrt{3}) (- 3 + \sqrt{3})}{6 - 4 \sqrt{3} + \sqrt{3}}$

Step 5.2.3.6

Add $- 4 \sqrt{3}$ and $\sqrt{3}$ .

$f (- 2 + \sqrt{3}) = \frac{(- 1 + \sqrt{3}) (- 3 + \sqrt{3})}{6 - 3 \sqrt{3}}$

Step 5.2.4

Expand $(- 1 + \sqrt{3}) (- 3 + \sqrt{3})$ using the FOIL Method.

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Step 5.2.4.1

Apply the distributive property.

$f (- 2 + \sqrt{3}) = \frac{- 1 (- 3 + \sqrt{3}) + \sqrt{3} (- 3 + \sqrt{3})}{6 - 3 \sqrt{3}}$

Step 5.2.4.2

Apply the distributive property.

$f (- 2 + \sqrt{3}) = \frac{- 1 \cdot - 3 - 1 \sqrt{3} + \sqrt{3} (- 3 + \sqrt{3})}{6 - 3 \sqrt{3}}$

Step 5.2.4.3

Apply the distributive property.

$f (- 2 + \sqrt{3}) = \frac{- 1 \cdot - 3 - 1 \sqrt{3} + \sqrt{3} \cdot - 3 + \sqrt{3} \sqrt{3}}{6 - 3 \sqrt{3}}$

Step 5.2.5

Simplify and combine like terms.

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Step 5.2.5.1

Simplify each term.

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Step 5.2.5.1.1

Multiply $- 1$ by $- 3$ .

$f (- 2 + \sqrt{3}) = \frac{3 - 1 \sqrt{3} + \sqrt{3} \cdot - 3 + \sqrt{3} \sqrt{3}}{6 - 3 \sqrt{3}}$

Step 5.2.5.1.2

Rewrite $- 1 \sqrt{3}$ as $- \sqrt{3}$ .

$f (- 2 + \sqrt{3}) = \frac{3 - \sqrt{3} + \sqrt{3} \cdot - 3 + \sqrt{3} \sqrt{3}}{6 - 3 \sqrt{3}}$

Step 5.2.5.1.3

Move $- 3$ to the left of $\sqrt{3}$ .

$f (- 2 + \sqrt{3}) = \frac{3 - \sqrt{3} - 3 \cdot \sqrt{3} + \sqrt{3} \sqrt{3}}{6 - 3 \sqrt{3}}$

Step 5.2.5.1.4

Combine using the product rule for radicals.

$f (- 2 + \sqrt{3}) = \frac{3 - \sqrt{3} - 3 \sqrt{3} + \sqrt{3 \cdot 3}}{6 - 3 \sqrt{3}}$

Step 5.2.5.1.5

Multiply $3$ by $3$ .

$f (- 2 + \sqrt{3}) = \frac{3 - \sqrt{3} - 3 \sqrt{3} + \sqrt{9}}{6 - 3 \sqrt{3}}$

Step 5.2.5.1.6

Rewrite $9$ as $3^{2}$ .

$f (- 2 + \sqrt{3}) = \frac{3 - \sqrt{3} - 3 \sqrt{3} + \sqrt{3^{2}}}{6 - 3 \sqrt{3}}$

Step 5.2.5.1.7

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

$f (- 2 + \sqrt{3}) = \frac{3 - \sqrt{3} - 3 \sqrt{3} + 3}{6 - 3 \sqrt{3}}$

Step 5.2.5.2

Add $3$ and $3$ .

$f (- 2 + \sqrt{3}) = \frac{6 - \sqrt{3} - 3 \sqrt{3}}{6 - 3 \sqrt{3}}$

Step 5.2.5.3

Subtract $3 \sqrt{3}$ from $- \sqrt{3}$ .

$f (- 2 + \sqrt{3}) = \frac{6 - 4 \sqrt{3}}{6 - 3 \sqrt{3}}$

Step 5.2.6

Multiply $\frac{6 - 4 \sqrt{3}}{6 - 3 \sqrt{3}}$ by $\frac{6 + 3 \sqrt{3}}{6 + 3 \sqrt{3}}$ .

$f (- 2 + \sqrt{3}) = \frac{6 - 4 \sqrt{3}}{6 - 3 \sqrt{3}} \cdot \frac{6 + 3 \sqrt{3}}{6 + 3 \sqrt{3}}$

Step 5.2.7

Simplify terms.

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Step 5.2.7.1

Multiply $\frac{6 - 4 \sqrt{3}}{6 - 3 \sqrt{3}}$ by $\frac{6 + 3 \sqrt{3}}{6 + 3 \sqrt{3}}$ .

$f (- 2 + \sqrt{3}) = \frac{(6 - 4 \sqrt{3}) (6 + 3 \sqrt{3})}{(6 - 3 \sqrt{3}) (6 + 3 \sqrt{3})}$

Step 5.2.7.2

Expand the denominator using the FOIL method.

$f (- 2 + \sqrt{3}) = \frac{(6 - 4 \sqrt{3}) (6 + 3 \sqrt{3})}{36 + 18 \sqrt{3} - 18 \sqrt{3} - 9 {\sqrt{3}}^{2}}$

Step 5.2.7.3

Simplify.

$f (- 2 + \sqrt{3}) = \frac{(6 - 4 \sqrt{3}) (6 + 3 \sqrt{3})}{9}$

Step 5.2.7.4

Cancel the common factor of $6 + 3 \sqrt{3}$ and $9$ .

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Step 5.2.7.4.1

Factor $3$ out of $(6 - 4 \sqrt{3}) (6 + 3 \sqrt{3})$ .

$f (- 2 + \sqrt{3}) = \frac{3 ((6 - 4 \sqrt{3}) (2 + \sqrt{3}))}{9}$

Step 5.2.7.4.2

Cancel the common factors.

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Step 5.2.7.4.2.1

Factor $3$ out of $9$ .

$f (- 2 + \sqrt{3}) = \frac{3 ((6 - 4 \sqrt{3}) (2 + \sqrt{3}))}{3 (3)}$

Step 5.2.7.4.2.2

Cancel the common factor.

$f (- 2 + \sqrt{3}) = \frac{3 ((6 - 4 \sqrt{3}) (2 + \sqrt{3}))}{3 \cdot 3}$

Step 5.2.7.4.2.3

Rewrite the expression.

$f (- 2 + \sqrt{3}) = \frac{(6 - 4 \sqrt{3}) (2 + \sqrt{3})}{3}$

Step 5.2.8

Simplify the numerator.

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Step 5.2.8.1

Expand $(6 - 4 \sqrt{3}) (2 + \sqrt{3})$ using the FOIL Method.

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Step 5.2.8.1.1

Apply the distributive property.

$f (- 2 + \sqrt{3}) = \frac{6 (2 + \sqrt{3}) - 4 \sqrt{3} (2 + \sqrt{3})}{3}$

Step 5.2.8.1.2

Apply the distributive property.

$f (- 2 + \sqrt{3}) = \frac{6 \cdot 2 + 6 \sqrt{3} - 4 \sqrt{3} (2 + \sqrt{3})}{3}$

Step 5.2.8.1.3

Apply the distributive property.

$f (- 2 + \sqrt{3}) = \frac{6 \cdot 2 + 6 \sqrt{3} - 4 \sqrt{3} \cdot 2 - 4 \sqrt{3} \sqrt{3}}{3}$

Step 5.2.8.2

Simplify and combine like terms.

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Step 5.2.8.2.1

Simplify each term.

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Step 5.2.8.2.1.1

Multiply $6$ by $2$ .

$f (- 2 + \sqrt{3}) = \frac{12 + 6 \sqrt{3} - 4 \sqrt{3} \cdot 2 - 4 \sqrt{3} \sqrt{3}}{3}$

Step 5.2.8.2.1.2

Multiply $2$ by $- 4$ .

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

Step 5.2.8.2.1.3

Multiply $- 4 \sqrt{3} \sqrt{3}$ .

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Step 5.2.8.2.1.3.1

Raise $\sqrt{3}$ to the power of $1$ .

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

Step 5.2.8.2.1.3.2

Raise $\sqrt{3}$ to the power of $1$ .

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

Step 5.2.8.2.1.3.3

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

$f (- 2 + \sqrt{3}) = \frac{12 + 6 \sqrt{3} - 8 \sqrt{3} - 4 {\sqrt{3}}^{1 + 1}}{3}$

Step 5.2.8.2.1.3.4

Add $1$ and $1$ .

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

Step 5.2.8.2.1.4

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 5.2.8.2.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$f (- 2 + \sqrt{3}) = \frac{12 + 6 \sqrt{3} - 8 \sqrt{3} - 4 {(3^{\frac{1}{2}})}^{2}}{3}$

Step 5.2.8.2.1.4.2

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

$f (- 2 + \sqrt{3}) = \frac{12 + 6 \sqrt{3} - 8 \sqrt{3} - 4 \cdot 3^{\frac{1}{2} \cdot 2}}{3}$

Step 5.2.8.2.1.4.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 5.2.8.2.1.4.4

Cancel the common factor of $2$ .

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Step 5.2.8.2.1.4.4.1

Cancel the common factor.

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

Step 5.2.8.2.1.4.4.2

Rewrite the expression.

$f (- 2 + \sqrt{3}) = \frac{12 + 6 \sqrt{3} - 8 \sqrt{3} - 4 \cdot 3}{3}$

Step 5.2.8.2.1.4.5

Evaluate the exponent.

$f (- 2 + \sqrt{3}) = \frac{12 + 6 \sqrt{3} - 8 \sqrt{3} - 4 \cdot 3}{3}$

Step 5.2.8.2.1.5

Multiply $- 4$ by $3$ .

$f (- 2 + \sqrt{3}) = \frac{12 + 6 \sqrt{3} - 8 \sqrt{3} - 12}{3}$

Step 5.2.8.2.2

Subtract $12$ from $12$ .

$f (- 2 + \sqrt{3}) = \frac{0 + 6 \sqrt{3} - 8 \sqrt{3}}{3}$

Step 5.2.8.2.3

Add $0$ and $6 \sqrt{3}$ .

$f (- 2 + \sqrt{3}) = \frac{6 \sqrt{3} - 8 \sqrt{3}}{3}$

Step 5.2.8.2.4

Subtract $8 \sqrt{3}$ from $6 \sqrt{3}$ .

$f (- 2 + \sqrt{3}) = \frac{- 2 \sqrt{3}}{3}$

Step 5.2.9

Move the negative in front of the fraction.

$f (- 2 + \sqrt{3}) = - \frac{2 \sqrt{3}}{3}$

Step 5.2.10

The final answer is $- \frac{2 \sqrt{3}}{3}$ .

$- \frac{2 \sqrt{3}}{3}$

Step 6

Solve the original function $f (x) = \frac{(x + 1) (x - 1)}{x^{2} + x + 1}$ at $x = - 2 - \sqrt{3}$ .

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Step 6.1

Replace the variable $x$ with $- 2 - \sqrt{3}$ in the expression.

$f (- 2 - \sqrt{3}) = \frac{((- 2 - \sqrt{3}) + 1) ((- 2 - \sqrt{3}) - 1)}{{(- 2 - \sqrt{3})}^{2} - 2 - \sqrt{3} + 1}$

Step 6.2

Simplify the result.

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Step 6.2.1

Remove parentheses.

$f (- 2 - \sqrt{3}) = \frac{((- 2 - \sqrt{3}) + 1) ((- 2 - \sqrt{3}) - 1)}{{(- 2 - \sqrt{3})}^{2} - 2 - \sqrt{3} + 1}$

Step 6.2.2

Simplify the numerator.

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Step 6.2.2.1

Add $- 2$ and $1$ .

$f (- 2 - \sqrt{3}) = \frac{(- 1 - \sqrt{3}) (- 2 - \sqrt{3} - 1)}{{(- 2 - \sqrt{3})}^{2} - 2 - \sqrt{3} + 1}$

Step 6.2.2.2

Subtract $1$ from $- 2$ .

$f (- 2 - \sqrt{3}) = \frac{(- 1 - \sqrt{3}) (- 3 - \sqrt{3})}{{(- 2 - \sqrt{3})}^{2} - 2 - \sqrt{3} + 1}$

Step 6.2.3

Simplify the denominator.

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Step 6.2.3.1

Rewrite ${(- 2 - \sqrt{3})}^{2}$ as $(- 2 - \sqrt{3}) (- 2 - \sqrt{3})$ .

$f (- 2 - \sqrt{3}) = \frac{(- 1 - \sqrt{3}) (- 3 - \sqrt{3})}{(- 2 - \sqrt{3}) (- 2 - \sqrt{3}) - 2 - \sqrt{3} + 1}$

Step 6.2.3.2

Expand $(- 2 - \sqrt{3}) (- 2 - \sqrt{3})$ using the FOIL Method.

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Step 6.2.3.2.1

Apply the distributive property.

$f (- 2 - \sqrt{3}) = \frac{(- 1 - \sqrt{3}) (- 3 - \sqrt{3})}{- 2 (- 2 - \sqrt{3}) - \sqrt{3} (- 2 - \sqrt{3}) - 2 - \sqrt{3} + 1}$

Step 6.2.3.2.2

Apply the distributive property.

$f (- 2 - \sqrt{3}) = \frac{(- 1 - \sqrt{3}) (- 3 - \sqrt{3})}{- 2 \cdot - 2 - 2 (- \sqrt{3}) - \sqrt{3} (- 2 - \sqrt{3}) - 2 - \sqrt{3} + 1}$

Step 6.2.3.2.3

Apply the distributive property.

$f (- 2 - \sqrt{3}) = \frac{(- 1 - \sqrt{3}) (- 3 - \sqrt{3})}{- 2 \cdot - 2 - 2 (- \sqrt{3}) - \sqrt{3} \cdot - 2 - \sqrt{3} (- \sqrt{3}) - 2 - \sqrt{3} + 1}$

Step 6.2.3.3

Simplify and combine like terms.

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Step 6.2.3.3.1

Simplify each term.

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Step 6.2.3.3.1.1

Multiply $- 2$ by $- 2$ .

$f (- 2 - \sqrt{3}) = \frac{(- 1 - \sqrt{3}) (- 3 - \sqrt{3})}{4 - 2 (- \sqrt{3}) - \sqrt{3} \cdot - 2 - \sqrt{3} (- \sqrt{3}) - 2 - \sqrt{3} + 1}$

Step 6.2.3.3.1.2

Multiply $- 1$ by $- 2$ .

$f (- 2 - \sqrt{3}) = \frac{(- 1 - \sqrt{3}) (- 3 - \sqrt{3})}{4 + 2 \sqrt{3} - \sqrt{3} \cdot - 2 - \sqrt{3} (- \sqrt{3}) - 2 - \sqrt{3} + 1}$

Step 6.2.3.3.1.3

Multiply $- 2$ by $- 1$ .

$f (- 2 - \sqrt{3}) = \frac{(- 1 - \sqrt{3}) (- 3 - \sqrt{3})}{4 + 2 \sqrt{3} + 2 \sqrt{3} - \sqrt{3} (- \sqrt{3}) - 2 - \sqrt{3} + 1}$

Step 6.2.3.3.1.4

Multiply $- \sqrt{3} (- \sqrt{3})$ .

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Step 6.2.3.3.1.4.1

Multiply $- 1$ by $- 1$ .

$f (- 2 - \sqrt{3}) = \frac{(- 1 - \sqrt{3}) (- 3 - \sqrt{3})}{4 + 2 \sqrt{3} + 2 \sqrt{3} + 1 \sqrt{3} \sqrt{3} - 2 - \sqrt{3} + 1}$

Step 6.2.3.3.1.4.2

Multiply $\sqrt{3}$ by $1$ .

$f (- 2 - \sqrt{3}) = \frac{(- 1 - \sqrt{3}) (- 3 - \sqrt{3})}{4 + 2 \sqrt{3} + 2 \sqrt{3} + \sqrt{3} \sqrt{3} - 2 - \sqrt{3} + 1}$

Step 6.2.3.3.1.4.3

Raise $\sqrt{3}$ to the power of $1$ .

$f (- 2 - \sqrt{3}) = \frac{(- 1 - \sqrt{3}) (- 3 - \sqrt{3})}{4 + 2 \sqrt{3} + 2 \sqrt{3} + \sqrt{3} \sqrt{3} - 2 - \sqrt{3} + 1}$

Step 6.2.3.3.1.4.4

Raise $\sqrt{3}$ to the power of $1$ .

$f (- 2 - \sqrt{3}) = \frac{(- 1 - \sqrt{3}) (- 3 - \sqrt{3})}{4 + 2 \sqrt{3} + 2 \sqrt{3} + \sqrt{3} \sqrt{3} - 2 - \sqrt{3} + 1}$

Step 6.2.3.3.1.4.5

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

$f (- 2 - \sqrt{3}) = \frac{(- 1 - \sqrt{3}) (- 3 - \sqrt{3})}{4 + 2 \sqrt{3} + 2 \sqrt{3} + {\sqrt{3}}^{1 + 1} - 2 - \sqrt{3} + 1}$

Step 6.2.3.3.1.4.6

Add $1$ and $1$ .

$f (- 2 - \sqrt{3}) = \frac{(- 1 - \sqrt{3}) (- 3 - \sqrt{3})}{4 + 2 \sqrt{3} + 2 \sqrt{3} + {\sqrt{3}}^{2} - 2 - \sqrt{3} + 1}$

Step 6.2.3.3.1.5

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 6.2.3.3.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$f (- 2 - \sqrt{3}) = \frac{(- 1 - \sqrt{3}) (- 3 - \sqrt{3})}{4 + 2 \sqrt{3} + 2 \sqrt{3} + {(3^{\frac{1}{2}})}^{2} - 2 - \sqrt{3} + 1}$

Step 6.2.3.3.1.5.2

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

$f (- 2 - \sqrt{3}) = \frac{(- 1 - \sqrt{3}) (- 3 - \sqrt{3})}{4 + 2 \sqrt{3} + 2 \sqrt{3} + 3^{\frac{1}{2} \cdot 2} - 2 - \sqrt{3} + 1}$

Step 6.2.3.3.1.5.3

Combine $\frac{1}{2}$ and $2$ .

$f (- 2 - \sqrt{3}) = \frac{(- 1 - \sqrt{3}) (- 3 - \sqrt{3})}{4 + 2 \sqrt{3} + 2 \sqrt{3} + 3^{\frac{2}{2}} - 2 - \sqrt{3} + 1}$

Step 6.2.3.3.1.5.4

Cancel the common factor of $2$ .

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Step 6.2.3.3.1.5.4.1

Cancel the common factor.

$f (- 2 - \sqrt{3}) = \frac{(- 1 - \sqrt{3}) (- 3 - \sqrt{3})}{4 + 2 \sqrt{3} + 2 \sqrt{3} + 3^{\frac{2}{2}} - 2 - \sqrt{3} + 1}$

Step 6.2.3.3.1.5.4.2

Rewrite the expression.

$f (- 2 - \sqrt{3}) = \frac{(- 1 - \sqrt{3}) (- 3 - \sqrt{3})}{4 + 2 \sqrt{3} + 2 \sqrt{3} + 3 - 2 - \sqrt{3} + 1}$

Step 6.2.3.3.1.5.5

Evaluate the exponent.

$f (- 2 - \sqrt{3}) = \frac{(- 1 - \sqrt{3}) (- 3 - \sqrt{3})}{4 + 2 \sqrt{3} + 2 \sqrt{3} + 3 - 2 - \sqrt{3} + 1}$

Step 6.2.3.3.2

Add $4$ and $3$ .

$f (- 2 - \sqrt{3}) = \frac{(- 1 - \sqrt{3}) (- 3 - \sqrt{3})}{7 + 2 \sqrt{3} + 2 \sqrt{3} - 2 - \sqrt{3} + 1}$

Step 6.2.3.3.3

Add $2 \sqrt{3}$ and $2 \sqrt{3}$ .

$f (- 2 - \sqrt{3}) = \frac{(- 1 - \sqrt{3}) (- 3 - \sqrt{3})}{7 + 4 \sqrt{3} - 2 - \sqrt{3} + 1}$

Step 6.2.3.4

Subtract $2$ from $7$ .

$f (- 2 - \sqrt{3}) = \frac{(- 1 - \sqrt{3}) (- 3 - \sqrt{3})}{5 + 4 \sqrt{3} - \sqrt{3} + 1}$

Step 6.2.3.5

Add $5$ and $1$ .

$f (- 2 - \sqrt{3}) = \frac{(- 1 - \sqrt{3}) (- 3 - \sqrt{3})}{6 + 4 \sqrt{3} - \sqrt{3}}$

Step 6.2.3.6

Subtract $\sqrt{3}$ from $4 \sqrt{3}$ .

$f (- 2 - \sqrt{3}) = \frac{(- 1 - \sqrt{3}) (- 3 - \sqrt{3})}{6 + 3 \sqrt{3}}$

Step 6.2.4

Expand $(- 1 - \sqrt{3}) (- 3 - \sqrt{3})$ using the FOIL Method.

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Step 6.2.4.1

Apply the distributive property.

$f (- 2 - \sqrt{3}) = \frac{- 1 (- 3 - \sqrt{3}) - \sqrt{3} (- 3 - \sqrt{3})}{6 + 3 \sqrt{3}}$

Step 6.2.4.2

Apply the distributive property.

$f (- 2 - \sqrt{3}) = \frac{- 1 \cdot - 3 - 1 (- \sqrt{3}) - \sqrt{3} (- 3 - \sqrt{3})}{6 + 3 \sqrt{3}}$

Step 6.2.4.3

Apply the distributive property.

$f (- 2 - \sqrt{3}) = \frac{- 1 \cdot - 3 - 1 (- \sqrt{3}) - \sqrt{3} \cdot - 3 - \sqrt{3} (- \sqrt{3})}{6 + 3 \sqrt{3}}$

Step 6.2.5

Simplify and combine like terms.

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Step 6.2.5.1

Simplify each term.

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Step 6.2.5.1.1

Multiply $- 1$ by $- 3$ .

$f (- 2 - \sqrt{3}) = \frac{3 - 1 (- \sqrt{3}) - \sqrt{3} \cdot - 3 - \sqrt{3} (- \sqrt{3})}{6 + 3 \sqrt{3}}$

Step 6.2.5.1.2

Multiply $- 1 (- \sqrt{3})$ .

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Step 6.2.5.1.2.1

Multiply $- 1$ by $- 1$ .

$f (- 2 - \sqrt{3}) = \frac{3 + 1 \sqrt{3} - \sqrt{3} \cdot - 3 - \sqrt{3} (- \sqrt{3})}{6 + 3 \sqrt{3}}$

Step 6.2.5.1.2.2

Multiply $\sqrt{3}$ by $1$ .

$f (- 2 - \sqrt{3}) = \frac{3 + \sqrt{3} - \sqrt{3} \cdot - 3 - \sqrt{3} (- \sqrt{3})}{6 + 3 \sqrt{3}}$

Step 6.2.5.1.3

Multiply $- 3$ by $- 1$ .

$f (- 2 - \sqrt{3}) = \frac{3 + \sqrt{3} + 3 \sqrt{3} - \sqrt{3} (- \sqrt{3})}{6 + 3 \sqrt{3}}$

Step 6.2.5.1.4

Multiply $- \sqrt{3} (- \sqrt{3})$ .

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Step 6.2.5.1.4.1

Multiply $- 1$ by $- 1$ .

$f (- 2 - \sqrt{3}) = \frac{3 + \sqrt{3} + 3 \sqrt{3} + 1 \sqrt{3} \sqrt{3}}{6 + 3 \sqrt{3}}$

Step 6.2.5.1.4.2

Multiply $\sqrt{3}$ by $1$ .

$f (- 2 - \sqrt{3}) = \frac{3 + \sqrt{3} + 3 \sqrt{3} + \sqrt{3} \sqrt{3}}{6 + 3 \sqrt{3}}$

Step 6.2.5.1.4.3

Raise $\sqrt{3}$ to the power of $1$ .

$f (- 2 - \sqrt{3}) = \frac{3 + \sqrt{3} + 3 \sqrt{3} + \sqrt{3} \sqrt{3}}{6 + 3 \sqrt{3}}$

Step 6.2.5.1.4.4

Raise $\sqrt{3}$ to the power of $1$ .

$f (- 2 - \sqrt{3}) = \frac{3 + \sqrt{3} + 3 \sqrt{3} + \sqrt{3} \sqrt{3}}{6 + 3 \sqrt{3}}$

Step 6.2.5.1.4.5

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

$f (- 2 - \sqrt{3}) = \frac{3 + \sqrt{3} + 3 \sqrt{3} + {\sqrt{3}}^{1 + 1}}{6 + 3 \sqrt{3}}$

Step 6.2.5.1.4.6

Add $1$ and $1$ .

$f (- 2 - \sqrt{3}) = \frac{3 + \sqrt{3} + 3 \sqrt{3} + {\sqrt{3}}^{2}}{6 + 3 \sqrt{3}}$

Step 6.2.5.1.5

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 6.2.5.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$f (- 2 - \sqrt{3}) = \frac{3 + \sqrt{3} + 3 \sqrt{3} + {(3^{\frac{1}{2}})}^{2}}{6 + 3 \sqrt{3}}$

Step 6.2.5.1.5.2

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

$f (- 2 - \sqrt{3}) = \frac{3 + \sqrt{3} + 3 \sqrt{3} + 3^{\frac{1}{2} \cdot 2}}{6 + 3 \sqrt{3}}$

Step 6.2.5.1.5.3

Combine $\frac{1}{2}$ and $2$ .

$f (- 2 - \sqrt{3}) = \frac{3 + \sqrt{3} + 3 \sqrt{3} + 3^{\frac{2}{2}}}{6 + 3 \sqrt{3}}$

Step 6.2.5.1.5.4

Cancel the common factor of $2$ .

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Step 6.2.5.1.5.4.1

Cancel the common factor.

$f (- 2 - \sqrt{3}) = \frac{3 + \sqrt{3} + 3 \sqrt{3} + 3^{\frac{2}{2}}}{6 + 3 \sqrt{3}}$

Step 6.2.5.1.5.4.2

Rewrite the expression.

$f (- 2 - \sqrt{3}) = \frac{3 + \sqrt{3} + 3 \sqrt{3} + 3}{6 + 3 \sqrt{3}}$

Step 6.2.5.1.5.5

Evaluate the exponent.

$f (- 2 - \sqrt{3}) = \frac{3 + \sqrt{3} + 3 \sqrt{3} + 3}{6 + 3 \sqrt{3}}$

Step 6.2.5.2

Add $3$ and $3$ .

$f (- 2 - \sqrt{3}) = \frac{6 + \sqrt{3} + 3 \sqrt{3}}{6 + 3 \sqrt{3}}$

Step 6.2.5.3

Add $\sqrt{3}$ and $3 \sqrt{3}$ .

$f (- 2 - \sqrt{3}) = \frac{6 + 4 \sqrt{3}}{6 + 3 \sqrt{3}}$

Step 6.2.6

Multiply $\frac{6 + 4 \sqrt{3}}{6 + 3 \sqrt{3}}$ by $\frac{6 - 3 \sqrt{3}}{6 - 3 \sqrt{3}}$ .

$f (- 2 - \sqrt{3}) = \frac{6 + 4 \sqrt{3}}{6 + 3 \sqrt{3}} \cdot \frac{6 - 3 \sqrt{3}}{6 - 3 \sqrt{3}}$

Step 6.2.7

Simplify terms.

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Step 6.2.7.1

Multiply $\frac{6 + 4 \sqrt{3}}{6 + 3 \sqrt{3}}$ by $\frac{6 - 3 \sqrt{3}}{6 - 3 \sqrt{3}}$ .

$f (- 2 - \sqrt{3}) = \frac{(6 + 4 \sqrt{3}) (6 - 3 \sqrt{3})}{(6 + 3 \sqrt{3}) (6 - 3 \sqrt{3})}$

Step 6.2.7.2

Expand the denominator using the FOIL method.

$f (- 2 - \sqrt{3}) = \frac{(6 + 4 \sqrt{3}) (6 - 3 \sqrt{3})}{36 - 18 \sqrt{3} + 18 \sqrt{3} - 9 {\sqrt{3}}^{2}}$

Step 6.2.7.3

Simplify.

$f (- 2 - \sqrt{3}) = \frac{(6 + 4 \sqrt{3}) (6 - 3 \sqrt{3})}{9}$

Step 6.2.7.4

Cancel the common factor of $6 - 3 \sqrt{3}$ and $9$ .

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Step 6.2.7.4.1

Factor $3$ out of $(6 + 4 \sqrt{3}) (6 - 3 \sqrt{3})$ .

$f (- 2 - \sqrt{3}) = \frac{3 ((6 + 4 \sqrt{3}) (2 - \sqrt{3}))}{9}$

Step 6.2.7.4.2

Cancel the common factors.

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Step 6.2.7.4.2.1

Factor $3$ out of $9$ .

$f (- 2 - \sqrt{3}) = \frac{3 ((6 + 4 \sqrt{3}) (2 - \sqrt{3}))}{3 (3)}$

Step 6.2.7.4.2.2

Cancel the common factor.

$f (- 2 - \sqrt{3}) = \frac{3 ((6 + 4 \sqrt{3}) (2 - \sqrt{3}))}{3 \cdot 3}$

Step 6.2.7.4.2.3

Rewrite the expression.

$f (- 2 - \sqrt{3}) = \frac{(6 + 4 \sqrt{3}) (2 - \sqrt{3})}{3}$

Step 6.2.8

Simplify the numerator.

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Step 6.2.8.1

Expand $(6 + 4 \sqrt{3}) (2 - \sqrt{3})$ using the FOIL Method.

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Step 6.2.8.1.1

Apply the distributive property.

$f (- 2 - \sqrt{3}) = \frac{6 (2 - \sqrt{3}) + 4 \sqrt{3} (2 - \sqrt{3})}{3}$

Step 6.2.8.1.2

Apply the distributive property.

$f (- 2 - \sqrt{3}) = \frac{6 \cdot 2 + 6 (- \sqrt{3}) + 4 \sqrt{3} (2 - \sqrt{3})}{3}$

Step 6.2.8.1.3

Apply the distributive property.

$f (- 2 - \sqrt{3}) = \frac{6 \cdot 2 + 6 (- \sqrt{3}) + 4 \sqrt{3} \cdot 2 + 4 \sqrt{3} (- \sqrt{3})}{3}$

Step 6.2.8.2

Simplify and combine like terms.

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Step 6.2.8.2.1

Simplify each term.

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Step 6.2.8.2.1.1

Multiply $6$ by $2$ .

$f (- 2 - \sqrt{3}) = \frac{12 + 6 (- \sqrt{3}) + 4 \sqrt{3} \cdot 2 + 4 \sqrt{3} (- \sqrt{3})}{3}$

Step 6.2.8.2.1.2

Multiply $- 1$ by $6$ .

$f (- 2 - \sqrt{3}) = \frac{12 - 6 \sqrt{3} + 4 \sqrt{3} \cdot 2 + 4 \sqrt{3} (- \sqrt{3})}{3}$

Step 6.2.8.2.1.3

Multiply $2$ by $4$ .

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

Step 6.2.8.2.1.4

Multiply $4 \sqrt{3} (- \sqrt{3})$ .

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Step 6.2.8.2.1.4.1

Multiply $- 1$ by $4$ .

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

Step 6.2.8.2.1.4.2

Raise $\sqrt{3}$ to the power of $1$ .

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

Step 6.2.8.2.1.4.3

Raise $\sqrt{3}$ to the power of $1$ .

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

Step 6.2.8.2.1.4.4

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

$f (- 2 - \sqrt{3}) = \frac{12 - 6 \sqrt{3} + 8 \sqrt{3} - 4 {\sqrt{3}}^{1 + 1}}{3}$

Step 6.2.8.2.1.4.5

Add $1$ and $1$ .

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

Step 6.2.8.2.1.5

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 6.2.8.2.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$f (- 2 - \sqrt{3}) = \frac{12 - 6 \sqrt{3} + 8 \sqrt{3} - 4 {(3^{\frac{1}{2}})}^{2}}{3}$

Step 6.2.8.2.1.5.2

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

$f (- 2 - \sqrt{3}) = \frac{12 - 6 \sqrt{3} + 8 \sqrt{3} - 4 \cdot 3^{\frac{1}{2} \cdot 2}}{3}$

Step 6.2.8.2.1.5.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 6.2.8.2.1.5.4

Cancel the common factor of $2$ .

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Step 6.2.8.2.1.5.4.1

Cancel the common factor.

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

Step 6.2.8.2.1.5.4.2

Rewrite the expression.

$f (- 2 - \sqrt{3}) = \frac{12 - 6 \sqrt{3} + 8 \sqrt{3} - 4 \cdot 3}{3}$

Step 6.2.8.2.1.5.5

Evaluate the exponent.

$f (- 2 - \sqrt{3}) = \frac{12 - 6 \sqrt{3} + 8 \sqrt{3} - 4 \cdot 3}{3}$

Step 6.2.8.2.1.6

Multiply $- 4$ by $3$ .

$f (- 2 - \sqrt{3}) = \frac{12 - 6 \sqrt{3} + 8 \sqrt{3} - 12}{3}$

Step 6.2.8.2.2

Subtract $12$ from $12$ .

$f (- 2 - \sqrt{3}) = \frac{0 - 6 \sqrt{3} + 8 \sqrt{3}}{3}$

Step 6.2.8.2.3

Subtract $6 \sqrt{3}$ from $0$ .

$f (- 2 - \sqrt{3}) = \frac{- 6 \sqrt{3} + 8 \sqrt{3}}{3}$

Step 6.2.8.2.4

Add $- 6 \sqrt{3}$ and $8 \sqrt{3}$ .

$f (- 2 - \sqrt{3}) = \frac{2 \sqrt{3}}{3}$

Step 6.2.9

The final answer is $\frac{2 \sqrt{3}}{3}$ .

$\frac{2 \sqrt{3}}{3}$

Step 7

The horizontal tangent lines on function $f (x) = \frac{(x + 1) (x - 1)}{x^{2} + x + 1}$ are $y = - \frac{2 \sqrt{3}}{3}, y = \frac{2 \sqrt{3}}{3}$ .

$y = - \frac{2 \sqrt{3}}{3}, y = \frac{2 \sqrt{3}}{3}$

Step 8