Calculus Examples

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

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

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

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

Step 1.1.2

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

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Step 1.1.2.1

Apply the distributive property.

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

Step 1.1.2.2

Apply the distributive property.

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

Step 1.1.2.3

Apply the distributive property.

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

Step 1.1.3

Simplify and combine like terms.

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Step 1.1.3.1

Simplify each term.

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Step 1.1.3.1.1

Multiply $x$ by $x$ .

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

Step 1.1.3.1.2

Multiply $x$ by $1$ .

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

Step 1.1.3.1.3

Multiply $x$ by $1$ .

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

Step 1.1.3.1.4

Multiply $1$ by $1$ .

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

Step 1.1.3.2

Add $x$ and $x$ .

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

Step 1.1.4

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{2} + 2 x + 1$ and $g (x) = 2 x - x^{2}$ .

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

Step 1.1.5

Differentiate.

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Step 1.1.5.1

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

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

Step 1.1.5.2

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

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

Step 1.1.5.3

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

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

Step 1.1.5.4

Multiply $2$ by $1$ .

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

Step 1.1.5.5

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

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

Step 1.1.5.6

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

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

Step 1.1.5.7

Multiply $2$ by $- 1$ .

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

Step 1.1.5.8

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

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

Step 1.1.5.9

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

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

Step 1.1.5.10

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

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

Step 1.1.5.11

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

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

Step 1.1.5.12

Multiply $2$ by $1$ .

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

Step 1.1.5.13

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

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

Step 1.1.5.14

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

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

Step 1.1.6

Simplify.

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Step 1.1.6.1

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

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Step 1.1.6.1.1

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

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

Step 1.1.6.1.2

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

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

Step 1.1.6.1.3

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

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

Step 1.1.6.2

Multiply $(x^{2} + 2 x + 1) (2 - 2 x) + (2 x - x^{2}) (2 x + 2)$ by $1$ .

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

Step 1.1.6.3

Reorder terms.

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

Step 1.1.6.4

Simplify each term.

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Step 1.1.6.4.1

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

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

Step 1.1.6.4.2

Simplify each term.

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Step 1.1.6.4.2.1

Multiply $2$ by $2$ .

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

Step 1.1.6.4.2.2

Multiply $2$ by $1$ .

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

Step 1.1.6.4.2.3

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

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Step 1.1.6.4.2.3.1

Move $x^{2}$ .

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

Step 1.1.6.4.2.3.2

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

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Step 1.1.6.4.2.3.2.1

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

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

Step 1.1.6.4.2.3.2.2

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

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

Step 1.1.6.4.2.3.3

Add $2$ and $1$ .

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

Step 1.1.6.4.2.4

Rewrite using the commutative property of multiplication.

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

Step 1.1.6.4.2.5

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

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Step 1.1.6.4.2.5.1

Move $x$ .

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

Step 1.1.6.4.2.5.2

Multiply $x$ by $x$ .

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

Step 1.1.6.4.2.6

Multiply $- 2$ by $2$ .

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

Step 1.1.6.4.2.7

Multiply $- 2$ by $1$ .

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

Step 1.1.6.4.3

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

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

Step 1.1.6.4.4

Subtract $2 x$ from $4 x$ .

$- 2 x^{2} + 2 x + 2 - 2 x^{3} + (2 x + 2) (2 x - x^{2})$

Step 1.1.6.4.5

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

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Step 1.1.6.4.5.1

Apply the distributive property.

$- 2 x^{2} + 2 x + 2 - 2 x^{3} + 2 x (2 x - x^{2}) + 2 (2 x - x^{2})$

Step 1.1.6.4.5.2

Apply the distributive property.

$- 2 x^{2} + 2 x + 2 - 2 x^{3} + 2 x (2 x) + 2 x (- x^{2}) + 2 (2 x - x^{2})$

Step 1.1.6.4.5.3

Apply the distributive property.

$- 2 x^{2} + 2 x + 2 - 2 x^{3} + 2 x (2 x) + 2 x (- x^{2}) + 2 (2 x) + 2 (- x^{2})$

Step 1.1.6.4.6

Simplify and combine like terms.

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Step 1.1.6.4.6.1

Simplify each term.

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Step 1.1.6.4.6.1.1

Rewrite using the commutative property of multiplication.

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

Step 1.1.6.4.6.1.2

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

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Step 1.1.6.4.6.1.2.1

Move $x$ .

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

Step 1.1.6.4.6.1.2.2

Multiply $x$ by $x$ .

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

Step 1.1.6.4.6.1.3

Multiply $2$ by $2$ .

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

Step 1.1.6.4.6.1.4

Rewrite using the commutative property of multiplication.

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

Step 1.1.6.4.6.1.5

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

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Step 1.1.6.4.6.1.5.1

Move $x^{2}$ .

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

Step 1.1.6.4.6.1.5.2

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

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Step 1.1.6.4.6.1.5.2.1

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

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

Step 1.1.6.4.6.1.5.2.2

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

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

Step 1.1.6.4.6.1.5.3

Add $2$ and $1$ .

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

Step 1.1.6.4.6.1.6

Multiply $2$ by $- 1$ .

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

Step 1.1.6.4.6.1.7

Multiply $2$ by $2$ .

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

Step 1.1.6.4.6.1.8

Multiply $- 1$ by $2$ .

$- 2 x^{2} + 2 x + 2 - 2 x^{3} + 4 x^{2} - 2 x^{3} + 4 x - 2 x^{2}$

Step 1.1.6.4.6.2

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

$- 2 x^{2} + 2 x + 2 - 2 x^{3} + 2 x^{2} - 2 x^{3} + 4 x$

Step 1.1.6.5

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

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Step 1.1.6.5.1

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

$2 x + 2 - 2 x^{3} + 0 - 2 x^{3} + 4 x$

Step 1.1.6.5.2

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

$2 x + 2 - 2 x^{3} - 2 x^{3} + 4 x$

Step 1.1.6.6

Add $2 x$ and $4 x$ .

$2 - 2 x^{3} - 2 x^{3} + 6 x$

Step 1.1.6.7

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

$f' (x) = 2 - 4 x^{3} + 6 x$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $2 - 4 x^{3} + 6 x$ .

$2 - 4 x^{3} + 6 x$

Step 2

Set the first derivative equal to $0$ then solve the equation $2 - 4 x^{3} + 6 x = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$2 - 4 x^{3} + 6 x = 0$

Step 2.2

Factor the left side of the equation.

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Step 2.2.1

Factor $- 2$ out of $2 - 4 x^{3} + 6 x$ .

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Step 2.2.1.1

Move $2$ .

$- 4 x^{3} + 6 x + 2 = 0$

Step 2.2.1.2

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

$- 2 (2 x^{3}) + 6 x + 2 = 0$

Step 2.2.1.3

Factor $- 2$ out of $6 x$ .

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

Step 2.2.1.4

Factor $- 2$ out of $2$ .

$- 2 (2 x^{3}) - 2 (- 3 x) - 2 \cdot - 1 = 0$

Step 2.2.1.5

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

$- 2 (2 x^{3} - 3 x) - 2 \cdot - 1 = 0$

Step 2.2.1.6

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

$- 2 (2 x^{3} - 3 x - 1) = 0$

Step 2.2.2

Factor.

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Step 2.2.2.1

Factor $2 x^{3} - 3 x - 1$ using the rational roots test.

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Step 2.2.2.1.1

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1$

$q = \pm 1, \pm 2$

Step 2.2.2.1.2

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 0.5$

Step 2.2.2.1.3

Substitute $- 1$ and simplify the expression. In this case, the expression is equal to $0$ so $- 1$ is a root of the polynomial.

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Step 2.2.2.1.3.1

Substitute $- 1$ into the polynomial.

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

Step 2.2.2.1.3.2

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

$2 \cdot - 1 - 3 \cdot - 1 - 1$

Step 2.2.2.1.3.3

Multiply $2$ by $- 1$ .

$- 2 - 3 \cdot - 1 - 1$

Step 2.2.2.1.3.4

Multiply $- 3$ by $- 1$ .

$- 2 + 3 - 1$

Step 2.2.2.1.3.5

Add $- 2$ and $3$ .

$1 - 1$

Step 2.2.2.1.3.6

Subtract $1$ from $1$ .

$0$

Step 2.2.2.1.4

Since $- 1$ is a known root, divide the polynomial by $x + 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

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

Step 2.2.2.1.5

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

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Step 2.2.2.1.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$1$

$2 x^{3}$

$0 x^{2}$

$3 x$

$1$

Step 2.2.2.1.5.2

Divide the highest order term in the dividend $2 x^{3}$ by the highest order term in divisor $x$ .

					$2 x^{2}$
	$x$	+	$1$		$2 x^{3}$	+	$0 x^{2}$	-	$3 x$	-	$1$

Step 2.2.2.1.5.3

Multiply the new quotient term by the divisor.

				$2 x^{2}$
$x$	+	$1$		$2 x^{3}$	+	$0 x^{2}$	-	$3 x$	-	$1$
			+	$2 x^{3}$	+	$2 x^{2}$

Step 2.2.2.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $2 x^{3} + 2 x^{2}$

				$2 x^{2}$
$x$	+	$1$		$2 x^{3}$	+	$0 x^{2}$	-	$3 x$	-	$1$
			-	$2 x^{3}$	-	$2 x^{2}$

Step 2.2.2.1.5.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$2 x^{2}$
$x$	+	$1$		$2 x^{3}$	+	$0 x^{2}$	-	$3 x$	-	$1$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$2 x^{2}$

Step 2.2.2.1.5.6

Pull the next terms from the original dividend down into the current dividend.

				$2 x^{2}$
$x$	+	$1$		$2 x^{3}$	+	$0 x^{2}$	-	$3 x$	-	$1$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$2 x^{2}$	-	$3 x$

Step 2.2.2.1.5.7

Divide the highest order term in the dividend $- 2 x^{2}$ by the highest order term in divisor $x$ .

				$2 x^{2}$	-	$2 x$
$x$	+	$1$		$2 x^{3}$	+	$0 x^{2}$	-	$3 x$	-	$1$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$2 x^{2}$	-	$3 x$

Step 2.2.2.1.5.8

Multiply the new quotient term by the divisor.

				$2 x^{2}$	-	$2 x$
$x$	+	$1$		$2 x^{3}$	+	$0 x^{2}$	-	$3 x$	-	$1$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$2 x^{2}$	-	$3 x$
					-	$2 x^{2}$	-	$2 x$

Step 2.2.2.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 2 x^{2} - 2 x$

				$2 x^{2}$	-	$2 x$
$x$	+	$1$		$2 x^{3}$	+	$0 x^{2}$	-	$3 x$	-	$1$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$2 x^{2}$	-	$3 x$
					+	$2 x^{2}$	+	$2 x$

Step 2.2.2.1.5.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$2 x^{2}$	-	$2 x$
$x$	+	$1$		$2 x^{3}$	+	$0 x^{2}$	-	$3 x$	-	$1$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$2 x^{2}$	-	$3 x$
					+	$2 x^{2}$	+	$2 x$
							-	$x$

Step 2.2.2.1.5.11

Pull the next terms from the original dividend down into the current dividend.

				$2 x^{2}$	-	$2 x$
$x$	+	$1$		$2 x^{3}$	+	$0 x^{2}$	-	$3 x$	-	$1$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$2 x^{2}$	-	$3 x$
					+	$2 x^{2}$	+	$2 x$
							-	$x$	-	$1$

Step 2.2.2.1.5.12

Divide the highest order term in the dividend $- x$ by the highest order term in divisor $x$ .

				$2 x^{2}$	-	$2 x$	-	$1$
$x$	+	$1$		$2 x^{3}$	+	$0 x^{2}$	-	$3 x$	-	$1$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$2 x^{2}$	-	$3 x$
					+	$2 x^{2}$	+	$2 x$
							-	$x$	-	$1$

Step 2.2.2.1.5.13

Multiply the new quotient term by the divisor.

				$2 x^{2}$	-	$2 x$	-	$1$
$x$	+	$1$		$2 x^{3}$	+	$0 x^{2}$	-	$3 x$	-	$1$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$2 x^{2}$	-	$3 x$
					+	$2 x^{2}$	+	$2 x$
							-	$x$	-	$1$
							-	$x$	-	$1$

Step 2.2.2.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- x - 1$

				$2 x^{2}$	-	$2 x$	-	$1$
$x$	+	$1$		$2 x^{3}$	+	$0 x^{2}$	-	$3 x$	-	$1$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$2 x^{2}$	-	$3 x$
					+	$2 x^{2}$	+	$2 x$
							-	$x$	-	$1$
							+	$x$	+	$1$

Step 2.2.2.1.5.15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$2 x^{2}$	-	$2 x$	-	$1$
$x$	+	$1$		$2 x^{3}$	+	$0 x^{2}$	-	$3 x$	-	$1$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$2 x^{2}$	-	$3 x$
					+	$2 x^{2}$	+	$2 x$
							-	$x$	-	$1$
							+	$x$	+	$1$
										$0$

Step 2.2.2.1.5.16

Since the remander is $0$ , the final answer is the quotient.

$2 x^{2} - 2 x - 1$

Step 2.2.2.1.6

Write $2 x^{3} - 3 x - 1$ as a set of factors.

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

Step 2.2.2.2

Remove unnecessary parentheses.

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

Step 2.3

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

$x + 1 = 0$

$2 x^{2} - 2 x - 1 = 0$

Step 2.4

Set $x + 1$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 2.4.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 2.5

Set $2 x^{2} - 2 x - 1$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $2 x^{2} - 2 x - 1$ equal to $0$ .

$2 x^{2} - 2 x - 1 = 0$

Step 2.5.2

Solve $2 x^{2} - 2 x - 1 = 0$ for $x$ .

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Step 2.5.2.1

Use the quadratic formula to find the solutions.

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

Step 2.5.2.2

Substitute the values $a = 2$ , $b = - 2$ , and $c = - 1$ into the quadratic formula and solve for $x$ .

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

Step 2.5.2.3

Simplify.

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Step 2.5.2.3.1

Simplify the numerator.

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Step 2.5.2.3.1.1

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

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

Step 2.5.2.3.1.2

Multiply $- 4 \cdot 2 \cdot - 1$ .

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Step 2.5.2.3.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{2 \pm \sqrt{4 - 8 \cdot - 1}}{2 \cdot 2}$

Step 2.5.2.3.1.2.2

Multiply $- 8$ by $- 1$ .

$x = \frac{2 \pm \sqrt{4 + 8}}{2 \cdot 2}$

Step 2.5.2.3.1.3

Add $4$ and $8$ .

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

Step 2.5.2.3.1.4

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

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Step 2.5.2.3.1.4.1

Factor $4$ out of $12$ .

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

Step 2.5.2.3.1.4.2

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

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

Step 2.5.2.3.1.5

Pull terms out from under the radical.

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

Step 2.5.2.3.2

Multiply $2$ by $2$ .

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

Step 2.5.2.3.3

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

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

Step 2.5.2.4

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

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Step 2.5.2.4.1

Simplify the numerator.

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Step 2.5.2.4.1.1

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

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

Step 2.5.2.4.1.2

Multiply $- 4 \cdot 2 \cdot - 1$ .

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Step 2.5.2.4.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{2 \pm \sqrt{4 - 8 \cdot - 1}}{2 \cdot 2}$

Step 2.5.2.4.1.2.2

Multiply $- 8$ by $- 1$ .

$x = \frac{2 \pm \sqrt{4 + 8}}{2 \cdot 2}$

Step 2.5.2.4.1.3

Add $4$ and $8$ .

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

Step 2.5.2.4.1.4

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

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Step 2.5.2.4.1.4.1

Factor $4$ out of $12$ .

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

Step 2.5.2.4.1.4.2

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

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

Step 2.5.2.4.1.5

Pull terms out from under the radical.

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

Step 2.5.2.4.2

Multiply $2$ by $2$ .

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

Step 2.5.2.4.3

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

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

Step 2.5.2.4.4

Change the $\pm$ to $+$ .

$x = \frac{1 + \sqrt{3}}{2}$

Step 2.5.2.5

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

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Step 2.5.2.5.1

Simplify the numerator.

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Step 2.5.2.5.1.1

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

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

Step 2.5.2.5.1.2

Multiply $- 4 \cdot 2 \cdot - 1$ .

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Step 2.5.2.5.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{2 \pm \sqrt{4 - 8 \cdot - 1}}{2 \cdot 2}$

Step 2.5.2.5.1.2.2

Multiply $- 8$ by $- 1$ .

$x = \frac{2 \pm \sqrt{4 + 8}}{2 \cdot 2}$

Step 2.5.2.5.1.3

Add $4$ and $8$ .

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

Step 2.5.2.5.1.4

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

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Step 2.5.2.5.1.4.1

Factor $4$ out of $12$ .

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

Step 2.5.2.5.1.4.2

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

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

Step 2.5.2.5.1.5

Pull terms out from under the radical.

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

Step 2.5.2.5.2

Multiply $2$ by $2$ .

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

Step 2.5.2.5.3

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

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

Step 2.5.2.5.4

Change the $\pm$ to $-$ .

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

Step 2.5.2.6

The final answer is the combination of both solutions.

$x = \frac{1 + \sqrt{3}}{2}, \frac{1 - \sqrt{3}}{2}$

Step 2.6

The final solution is all the values that make $- 2 (x + 1) (2 x^{2} - 2 x - 1) = 0$ true.

$x = - 1, \frac{1 + \sqrt{3}}{2}, \frac{1 - \sqrt{3}}{2}$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 4

Evaluate ${(x + 1)}^{2} (2 x - x^{2})$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = - 1$ .

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Step 4.1.1

Substitute $- 1$ for $x$ .

${((- 1) + 1)}^{2} (2 (- 1) - {(- 1)}^{2})$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Simplify the expression.

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Step 4.1.2.1.1

Add $- 1$ and $1$ .

$0^{2} (2 (- 1) - {(- 1)}^{2})$

Step 4.1.2.1.2

Raising $0$ to any positive power yields $0$ .

$0 (2 (- 1) - {(- 1)}^{2})$

Step 4.1.2.2

Simplify each term.

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Step 4.1.2.2.1

Multiply $2$ by $- 1$ .

$0 (- 2 - {(- 1)}^{2})$

Step 4.1.2.2.2

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Step 4.1.2.2.2.1

Multiply $- 1$ by ${(- 1)}^{2}$ .

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Step 4.1.2.2.2.1.1

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

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

Step 4.1.2.2.2.1.2

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

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

Step 4.1.2.2.2.2

Add $1$ and $2$ .

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

Step 4.1.2.2.3

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

$0 (- 2 - 1)$

Step 4.1.2.3

Simplify the expression.

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Step 4.1.2.3.1

Subtract $1$ from $- 2$ .

$0 \cdot - 3$

Step 4.1.2.3.2

Multiply $0$ by $- 3$ .

$0$

Step 4.2

Evaluate at $x = \frac{1 + \sqrt{3}}{2}$ .

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Step 4.2.1

Substitute $\frac{1 + \sqrt{3}}{2}$ for $x$ .

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

Step 4.2.2

Simplify.

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Step 4.2.2.1

Simplify the expression.

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Step 4.2.2.1.1

Write $1$ as a fraction with a common denominator.

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

Step 4.2.2.1.2

Combine the numerators over the common denominator.

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

Step 4.2.2.1.3

Add $1$ and $2$ .

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

Step 4.2.2.1.4

Apply the product rule to $\frac{3 + \sqrt{3}}{2}$ .

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

Step 4.2.2.1.5

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

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

Step 4.2.2.1.6

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

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

Step 4.2.2.2

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

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Step 4.2.2.2.1

Apply the distributive property.

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

Step 4.2.2.2.2

Apply the distributive property.

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

Step 4.2.2.2.3

Apply the distributive property.

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

Step 4.2.2.3

Simplify and combine like terms.

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Step 4.2.2.3.1

Simplify each term.

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Step 4.2.2.3.1.1

Multiply $3$ by $3$ .

$\frac{9 + 3 \sqrt{3} + \sqrt{3} \cdot 3 + \sqrt{3} \sqrt{3}}{4} (2 (\frac{1 + \sqrt{3}}{2}) - {(\frac{1 + \sqrt{3}}{2})}^{2})$

Step 4.2.2.3.1.2

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

$\frac{9 + 3 \sqrt{3} + 3 \cdot \sqrt{3} + \sqrt{3} \sqrt{3}}{4} (2 (\frac{1 + \sqrt{3}}{2}) - {(\frac{1 + \sqrt{3}}{2})}^{2})$

Step 4.2.2.3.1.3

Combine using the product rule for radicals.

$\frac{9 + 3 \sqrt{3} + 3 \sqrt{3} + \sqrt{3 \cdot 3}}{4} (2 (\frac{1 + \sqrt{3}}{2}) - {(\frac{1 + \sqrt{3}}{2})}^{2})$

Step 4.2.2.3.1.4

Multiply $3$ by $3$ .

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

Step 4.2.2.3.1.5

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

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

Step 4.2.2.3.1.6

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

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

Step 4.2.2.3.2

Add $9$ and $3$ .

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

Step 4.2.2.3.3

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

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

Step 4.2.2.4

Cancel the common factor of $12 + 6 \sqrt{3}$ and $4$ .

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Step 4.2.2.4.1

Factor $2$ out of $12$ .

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

Step 4.2.2.4.2

Factor $2$ out of $6 \sqrt{3}$ .

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

Step 4.2.2.4.3

Factor $2$ out of $2 (6) + 2 (3 \sqrt{3})$ .

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

Step 4.2.2.4.4

Cancel the common factors.

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Step 4.2.2.4.4.1

Factor $2$ out of $4$ .

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

Step 4.2.2.4.4.2

Cancel the common factor.

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

Step 4.2.2.4.4.3

Rewrite the expression.

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

Step 4.2.2.5

Simplify each term.

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Step 4.2.2.5.1

Cancel the common factor of $2$ .

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Step 4.2.2.5.1.1

Cancel the common factor.

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

Step 4.2.2.5.1.2

Rewrite the expression.

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

Step 4.2.2.5.2

Apply the product rule to $\frac{1 + \sqrt{3}}{2}$ .

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

Step 4.2.2.5.3

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

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

Step 4.2.2.5.4

Rewrite ${(1 + \sqrt{3})}^{2}$ as $(1 + \sqrt{3}) (1 + \sqrt{3})$ .

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

Step 4.2.2.5.5

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

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Step 4.2.2.5.5.1

Apply the distributive property.

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

Step 4.2.2.5.5.2

Apply the distributive property.

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

Step 4.2.2.5.5.3

Apply the distributive property.

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

Step 4.2.2.5.6

Simplify and combine like terms.

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Step 4.2.2.5.6.1

Simplify each term.

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Step 4.2.2.5.6.1.1

Multiply $1$ by $1$ .

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

Step 4.2.2.5.6.1.2

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

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

Step 4.2.2.5.6.1.3

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

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

Step 4.2.2.5.6.1.4

Combine using the product rule for radicals.

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

Step 4.2.2.5.6.1.5

Multiply $3$ by $3$ .

$\frac{6 + 3 \sqrt{3}}{2} (1 + \sqrt{3} - \frac{1 + \sqrt{3} + \sqrt{3} + \sqrt{9}}{4})$

Step 4.2.2.5.6.1.6

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

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

Step 4.2.2.5.6.1.7

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

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

Step 4.2.2.5.6.2

Add $1$ and $3$ .

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

Step 4.2.2.5.6.3

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

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

Step 4.2.2.5.7

Cancel the common factor of $4 + 2 \sqrt{3}$ and $4$ .

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Step 4.2.2.5.7.1

Factor $2$ out of $4$ .

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

Step 4.2.2.5.7.2

Factor $2$ out of $2 \sqrt{3}$ .

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

Step 4.2.2.5.7.3

Factor $2$ out of $2 \cdot 2 + 2 (\sqrt{3})$ .

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

Step 4.2.2.5.7.4

Cancel the common factors.

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Step 4.2.2.5.7.4.1

Factor $2$ out of $4$ .

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

Step 4.2.2.5.7.4.2

Cancel the common factor.

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

Step 4.2.2.5.7.4.3

Rewrite the expression.

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

Step 4.2.2.6

Simplify the expression.

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Step 4.2.2.6.1

Write $1$ as a fraction with a common denominator.

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

Step 4.2.2.6.2

Combine the numerators over the common denominator.

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

Step 4.2.2.7

Simplify each term.

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Step 4.2.2.7.1

Simplify the numerator.

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Step 4.2.2.7.1.1

Apply the distributive property.

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

Step 4.2.2.7.1.2

Multiply $- 1$ by $2$ .

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

Step 4.2.2.7.1.3

Subtract $2$ from $2$ .

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

Step 4.2.2.7.1.4

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

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

Step 4.2.2.7.2

Move the negative in front of the fraction.

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

Step 4.2.2.8

To write $\sqrt{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 4.2.2.9

Combine fractions.

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Step 4.2.2.9.1

Combine $\sqrt{3}$ and $\frac{2}{2}$ .

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

Step 4.2.2.9.2

Combine the numerators over the common denominator.

$\frac{6 + 3 \sqrt{3}}{2} \cdot \frac{- \sqrt{3} + \sqrt{3} \cdot 2}{2}$

Step 4.2.2.10

Simplify the numerator.

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Step 4.2.2.10.1

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

$\frac{6 + 3 \sqrt{3}}{2} \cdot \frac{- \sqrt{3} + 2 \cdot \sqrt{3}}{2}$

Step 4.2.2.10.2

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

$\frac{6 + 3 \sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}$

Step 4.2.2.11

Multiply $\frac{6 + 3 \sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}$ .

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Step 4.2.2.11.1

Multiply $\frac{6 + 3 \sqrt{3}}{2}$ by $\frac{\sqrt{3}}{2}$ .

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

Step 4.2.2.11.2

Multiply $2$ by $2$ .

$\frac{(6 + 3 \sqrt{3}) \sqrt{3}}{4}$

Step 4.2.2.12

Apply the distributive property.

$\frac{6 \sqrt{3} + 3 \sqrt{3} \sqrt{3}}{4}$

Step 4.2.2.13

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

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Step 4.2.2.13.1

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

$\frac{6 \sqrt{3} + 3 ({\sqrt{3}}^{1} \sqrt{3})}{4}$

Step 4.2.2.13.2

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

$\frac{6 \sqrt{3} + 3 ({\sqrt{3}}^{1} {\sqrt{3}}^{1})}{4}$

Step 4.2.2.13.3

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

$\frac{6 \sqrt{3} + 3 {\sqrt{3}}^{1 + 1}}{4}$

Step 4.2.2.13.4

Add $1$ and $1$ .

$\frac{6 \sqrt{3} + 3 {\sqrt{3}}^{2}}{4}$

Step 4.2.2.14

Simplify each term.

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Step 4.2.2.14.1

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

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Step 4.2.2.14.1.1

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

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

Step 4.2.2.14.1.2

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

$\frac{6 \sqrt{3} + 3 \cdot 3^{\frac{1}{2} \cdot 2}}{4}$

Step 4.2.2.14.1.3

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

$\frac{6 \sqrt{3} + 3 \cdot 3^{\frac{2}{2}}}{4}$

Step 4.2.2.14.1.4

Cancel the common factor of $2$ .

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Step 4.2.2.14.1.4.1

Cancel the common factor.

$\frac{6 \sqrt{3} + 3 \cdot 3^{\frac{2}{2}}}{4}$

Step 4.2.2.14.1.4.2

Rewrite the expression.

$\frac{6 \sqrt{3} + 3 \cdot 3^{1}}{4}$

Step 4.2.2.14.1.5

Evaluate the exponent.

$\frac{6 \sqrt{3} + 3 \cdot 3}{4}$

Step 4.2.2.14.2

Multiply $3$ by $3$ .

$\frac{6 \sqrt{3} + 9}{4}$

Step 4.3

Evaluate at $x = \frac{1 - \sqrt{3}}{2}$ .

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Step 4.3.1

Substitute $\frac{1 - \sqrt{3}}{2}$ for $x$ .

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

Step 4.3.2

Simplify.

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Step 4.3.2.1

Simplify the expression.

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Step 4.3.2.1.1

Write $1$ as a fraction with a common denominator.

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

Step 4.3.2.1.2

Combine the numerators over the common denominator.

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

Step 4.3.2.1.3

Add $1$ and $2$ .

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

Step 4.3.2.1.4

Apply the product rule to $\frac{3 - \sqrt{3}}{2}$ .

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

Step 4.3.2.1.5

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

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

Step 4.3.2.1.6

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

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

Step 4.3.2.2

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

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Step 4.3.2.2.1

Apply the distributive property.

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

Step 4.3.2.2.2

Apply the distributive property.

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

Step 4.3.2.2.3

Apply the distributive property.

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

Step 4.3.2.3

Simplify and combine like terms.

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Step 4.3.2.3.1

Simplify each term.

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Step 4.3.2.3.1.1

Multiply $3$ by $3$ .

$\frac{9 + 3 (- \sqrt{3}) - \sqrt{3} \cdot 3 - \sqrt{3} (- \sqrt{3})}{4} (2 (\frac{1 - \sqrt{3}}{2}) - {(\frac{1 - \sqrt{3}}{2})}^{2})$

Step 4.3.2.3.1.2

Multiply $- 1$ by $3$ .

$\frac{9 - 3 \sqrt{3} - \sqrt{3} \cdot 3 - \sqrt{3} (- \sqrt{3})}{4} (2 (\frac{1 - \sqrt{3}}{2}) - {(\frac{1 - \sqrt{3}}{2})}^{2})$

Step 4.3.2.3.1.3

Multiply $3$ by $- 1$ .

$\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} - \sqrt{3} (- \sqrt{3})}{4} (2 (\frac{1 - \sqrt{3}}{2}) - {(\frac{1 - \sqrt{3}}{2})}^{2})$

Step 4.3.2.3.1.4

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

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Step 4.3.2.3.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 4.3.2.3.1.4.2

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

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

Step 4.3.2.3.1.4.3

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

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

Step 4.3.2.3.1.4.4

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

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

Step 4.3.2.3.1.4.5

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

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

Step 4.3.2.3.1.4.6

Add $1$ and $1$ .

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

Step 4.3.2.3.1.5

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

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Step 4.3.2.3.1.5.1

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

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

Step 4.3.2.3.1.5.2

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

$\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{\frac{1}{2} \cdot 2}}{4} (2 (\frac{1 - \sqrt{3}}{2}) - {(\frac{1 - \sqrt{3}}{2})}^{2})$

Step 4.3.2.3.1.5.3

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

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

Step 4.3.2.3.1.5.4

Cancel the common factor of $2$ .

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Step 4.3.2.3.1.5.4.1

Cancel the common factor.

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

Step 4.3.2.3.1.5.4.2

Rewrite the expression.

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

Step 4.3.2.3.1.5.5

Evaluate the exponent.

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

Step 4.3.2.3.2

Add $9$ and $3$ .

$\frac{12 - 3 \sqrt{3} - 3 \sqrt{3}}{4} (2 (\frac{1 - \sqrt{3}}{2}) - {(\frac{1 - \sqrt{3}}{2})}^{2})$

Step 4.3.2.3.3

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

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

Step 4.3.2.4

Simplify terms.

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Step 4.3.2.4.1

Cancel the common factor of $12 - 6 \sqrt{3}$ and $4$ .

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Step 4.3.2.4.1.1

Factor $2$ out of $12$ .

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

Step 4.3.2.4.1.2

Factor $2$ out of $- 6 \sqrt{3}$ .

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

Step 4.3.2.4.1.3

Factor $2$ out of $2 (6) + 2 (- 3 \sqrt{3})$ .

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

Step 4.3.2.4.1.4

Cancel the common factors.

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Step 4.3.2.4.1.4.1

Factor $2$ out of $4$ .

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

Step 4.3.2.4.1.4.2

Cancel the common factor.

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

Step 4.3.2.4.1.4.3

Rewrite the expression.

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

Step 4.3.2.4.2

Simplify each term.

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Step 4.3.2.4.2.1

Cancel the common factor of $2$ .

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Step 4.3.2.4.2.1.1

Cancel the common factor.

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

Step 4.3.2.4.2.1.2

Rewrite the expression.

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

Step 4.3.2.4.2.2

Apply the product rule to $\frac{1 - \sqrt{3}}{2}$ .

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

Step 4.3.2.4.2.3

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

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

Step 4.3.2.4.2.4

Rewrite ${(1 - \sqrt{3})}^{2}$ as $(1 - \sqrt{3}) (1 - \sqrt{3})$ .

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

Step 4.3.2.4.2.5

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

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Step 4.3.2.4.2.5.1

Apply the distributive property.

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

Step 4.3.2.4.2.5.2

Apply the distributive property.

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

Step 4.3.2.4.2.5.3

Apply the distributive property.

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

Step 4.3.2.4.2.6

Simplify and combine like terms.

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Step 4.3.2.4.2.6.1

Simplify each term.

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Step 4.3.2.4.2.6.1.1

Multiply $1$ by $1$ .

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

Step 4.3.2.4.2.6.1.2

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

$\frac{6 - 3 \sqrt{3}}{2} (1 - \sqrt{3} - \frac{1 - \sqrt{3} - \sqrt{3} \cdot 1 - \sqrt{3} (- \sqrt{3})}{4})$

Step 4.3.2.4.2.6.1.3

Multiply $- 1$ by $1$ .

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

Step 4.3.2.4.2.6.1.4

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

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Step 4.3.2.4.2.6.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 4.3.2.4.2.6.1.4.2

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

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

Step 4.3.2.4.2.6.1.4.3

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

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

Step 4.3.2.4.2.6.1.4.4

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

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

Step 4.3.2.4.2.6.1.4.5

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

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

Step 4.3.2.4.2.6.1.4.6

Add $1$ and $1$ .

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

Step 4.3.2.4.2.6.1.5

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

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Step 4.3.2.4.2.6.1.5.1

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

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

Step 4.3.2.4.2.6.1.5.2

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

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

Step 4.3.2.4.2.6.1.5.3

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

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

Step 4.3.2.4.2.6.1.5.4

Cancel the common factor of $2$ .

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Step 4.3.2.4.2.6.1.5.4.1

Cancel the common factor.

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

Step 4.3.2.4.2.6.1.5.4.2

Rewrite the expression.

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

Step 4.3.2.4.2.6.1.5.5

Evaluate the exponent.

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

Step 4.3.2.4.2.6.2

Add $1$ and $3$ .

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

Step 4.3.2.4.2.6.3

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

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

Step 4.3.2.4.2.7

Cancel the common factor of $4 - 2 \sqrt{3}$ and $4$ .

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Step 4.3.2.4.2.7.1

Factor $2$ out of $4$ .

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

Step 4.3.2.4.2.7.2

Factor $2$ out of $- 2 \sqrt{3}$ .

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

Step 4.3.2.4.2.7.3

Factor $2$ out of $2 (2) + 2 (- \sqrt{3})$ .

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

Step 4.3.2.4.2.7.4

Cancel the common factors.

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Step 4.3.2.4.2.7.4.1

Factor $2$ out of $4$ .

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

Step 4.3.2.4.2.7.4.2

Cancel the common factor.

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

Step 4.3.2.4.2.7.4.3

Rewrite the expression.

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

Step 4.3.2.4.3

Simplify the expression.

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Step 4.3.2.4.3.1

Write $1$ as a fraction with a common denominator.

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

Step 4.3.2.4.3.2

Combine the numerators over the common denominator.

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

Step 4.3.2.5

Simplify the numerator.

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Step 4.3.2.5.1

Apply the distributive property.

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

Step 4.3.2.5.2

Multiply $- 1$ by $2$ .

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

Step 4.3.2.5.3

Multiply $- - \sqrt{3}$ .

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Step 4.3.2.5.3.1

Multiply $- 1$ by $- 1$ .

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

Step 4.3.2.5.3.2

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

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

Step 4.3.2.5.4

Subtract $2$ from $2$ .

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

Step 4.3.2.5.5

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

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

Step 4.3.2.6

To write $- \sqrt{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 4.3.2.7

Combine fractions.

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Step 4.3.2.7.1

Combine $- \sqrt{3}$ and $\frac{2}{2}$ .

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

Step 4.3.2.7.2

Combine the numerators over the common denominator.

$\frac{6 - 3 \sqrt{3}}{2} \cdot \frac{\sqrt{3} - \sqrt{3} \cdot 2}{2}$

Step 4.3.2.8

Simplify the numerator.

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Step 4.3.2.8.1

Multiply $2$ by $- 1$ .

$\frac{6 - 3 \sqrt{3}}{2} \cdot \frac{\sqrt{3} - 2 \sqrt{3}}{2}$

Step 4.3.2.8.2

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

$\frac{6 - 3 \sqrt{3}}{2} \cdot \frac{- \sqrt{3}}{2}$

Step 4.3.2.9

Move the negative in front of the fraction.

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

Step 4.3.2.10

Multiply $\frac{6 - 3 \sqrt{3}}{2} (- \frac{\sqrt{3}}{2})$ .

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Step 4.3.2.10.1

Multiply $\frac{6 - 3 \sqrt{3}}{2}$ by $\frac{\sqrt{3}}{2}$ .

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

Step 4.3.2.10.2

Multiply $2$ by $2$ .

$- \frac{(6 - 3 \sqrt{3}) \sqrt{3}}{4}$

Step 4.3.2.11

Apply the distributive property.

$- \frac{6 \sqrt{3} - 3 \sqrt{3} \sqrt{3}}{4}$

Step 4.3.2.12

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

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Step 4.3.2.12.1

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

$- \frac{6 \sqrt{3} - 3 ({\sqrt{3}}^{1} \sqrt{3})}{4}$

Step 4.3.2.12.2

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

$- \frac{6 \sqrt{3} - 3 ({\sqrt{3}}^{1} {\sqrt{3}}^{1})}{4}$

Step 4.3.2.12.3

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

$- \frac{6 \sqrt{3} - 3 {\sqrt{3}}^{1 + 1}}{4}$

Step 4.3.2.12.4

Add $1$ and $1$ .

$- \frac{6 \sqrt{3} - 3 {\sqrt{3}}^{2}}{4}$

Step 4.3.2.13

Simplify each term.

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Step 4.3.2.13.1

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

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Step 4.3.2.13.1.1

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

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

Step 4.3.2.13.1.2

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

$- \frac{6 \sqrt{3} - 3 \cdot 3^{\frac{1}{2} \cdot 2}}{4}$

Step 4.3.2.13.1.3

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

$- \frac{6 \sqrt{3} - 3 \cdot 3^{\frac{2}{2}}}{4}$

Step 4.3.2.13.1.4

Cancel the common factor of $2$ .

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Step 4.3.2.13.1.4.1

Cancel the common factor.

$- \frac{6 \sqrt{3} - 3 \cdot 3^{\frac{2}{2}}}{4}$

Step 4.3.2.13.1.4.2

Rewrite the expression.

$- \frac{6 \sqrt{3} - 3 \cdot 3^{1}}{4}$

Step 4.3.2.13.1.5

Evaluate the exponent.

$- \frac{6 \sqrt{3} - 3 \cdot 3}{4}$

Step 4.3.2.13.2

Multiply $- 3$ by $3$ .

$- \frac{6 \sqrt{3} - 9}{4}$

Step 4.4

List all of the points.

$(- 1, 0), (\frac{1 + \sqrt{3}}{2}, \frac{6 \sqrt{3} + 9}{4}), (\frac{1 - \sqrt{3}}{2}, - \frac{6 \sqrt{3} - 9}{4})$

Step 5