Algebra Examples

$- \frac{1}{4}$ , $\sqrt{7}$ , $- 6 i$

Step 1

Roots are the points where the graph intercepts with the x-axis $(y = 0)$ .

$y = 0$ at the roots

Step 2

The root at $x = - \frac{1}{4}$ was found by solving for $x$ when $x - (- \frac{1}{4}) = y$ and $y = 0$ .

The factor is $x + \frac{1}{4}$

Step 3

The root at $x = \sqrt{7}$ was found by solving for $x$ when $x - (\sqrt{7}) = y$ and $y = 0$ .

The factor is $x - \sqrt{7}$

Step 4

The root at $x = - 6 i$ was found by solving for $x$ when $x - (- 6 i) = y$ and $y = 0$ .

The factor is $x + 6 i$

Step 5

The root at $x = 6 i$ was found by solving for $x$ when $x - (6 i) = y$ and $y = 0$ .

The factor is $x - 6 i$

Step 6

Combine all the factors into a single equation.

$y = (x + \frac{1}{4}) (x - \sqrt{7}) (x + 6 i) (x - 6 i)$

Step 7

Multiply all the factors to simplify the equation $y = (x + \frac{1}{4}) (x - \sqrt{7}) (x + 6 i) (x - 6 i)$ .

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Step 7.1

Expand $(x + \frac{1}{4}) (x - \sqrt{7})$ using the FOIL Method.

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Step 7.1.1

Apply the distributive property.

$y = (x (x - \sqrt{7}) + \frac{1}{4} \cdot (x - \sqrt{7})) (x + 6 i) (x - 6 i)$

Step 7.1.2

Apply the distributive property.

$y = (x \cdot x + x (- \sqrt{7}) + \frac{1}{4} \cdot (x - \sqrt{7})) (x + 6 i) (x - 6 i)$

Step 7.1.3

Apply the distributive property.

$y = (x \cdot x + x (- \sqrt{7}) + \frac{1}{4} x + \frac{1}{4} \cdot (- \sqrt{7})) (x + 6 i) (x - 6 i)$

Step 7.2

Simplify and combine like terms.

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Step 7.2.1

Simplify each term.

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Step 7.2.1.1

Multiply $x$ by $x$ .

$y = (x^{2} + x (- \sqrt{7}) + \frac{1}{4} x + \frac{1}{4} \cdot (- \sqrt{7})) (x + 6 i) (x - 6 i)$

Step 7.2.1.2

Combine $\frac{1}{4}$ and $x$ .

$y = (x^{2} + x (- \sqrt{7}) + \frac{x}{4} + \frac{1}{4} \cdot (- \sqrt{7})) (x + 6 i) (x - 6 i)$

Step 7.2.1.3

Combine $\frac{1}{4}$ and $\sqrt{7}$ .

$y = (x^{2} + x (- \sqrt{7}) + \frac{x}{4} - \frac{\sqrt{7}}{4}) (x + 6 i) (x - 6 i)$

Step 7.2.2

Subtract $\frac{\sqrt{7}}{4}$ from $x (- \sqrt{7})$ .

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Step 7.2.2.1

Reorder $x$ and $- 1$ .

$y = (x^{2} + \frac{x}{4} - 1 \cdot (x \sqrt{7}) - \frac{\sqrt{7}}{4}) (x + 6 i) (x - 6 i)$

Step 7.2.2.2

To write $- 1 \cdot x \sqrt{7}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$y = (x^{2} + \frac{x}{4} - 1 \cdot (x \sqrt{7}) \cdot \frac{4}{4} - \frac{\sqrt{7}}{4}) (x + 6 i) (x - 6 i)$

Step 7.2.2.3

Combine $- 1 \cdot x \sqrt{7}$ and $\frac{4}{4}$ .

$y = (x^{2} + \frac{x}{4} + \frac{- 1 \cdot (x \sqrt{7}) \cdot 4}{4} - \frac{\sqrt{7}}{4}) (x + 6 i) (x - 6 i)$

Step 7.2.2.4

Combine the numerators over the common denominator.

$y = (x^{2} + \frac{x}{4} + \frac{- 1 \cdot (x \sqrt{7}) \cdot 4 - \sqrt{7}}{4}) (x + 6 i) (x - 6 i)$

Step 7.2.3

To write $x^{2}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$y = (\frac{x}{4} + x^{2} \cdot \frac{4}{4} + \frac{- 1 \cdot (x \sqrt{7}) \cdot 4 - \sqrt{7}}{4}) (x + 6 i) (x - 6 i)$

Step 7.2.4

Combine $x^{2}$ and $\frac{4}{4}$ .

$y = (\frac{x}{4} + \frac{x^{2} \cdot 4}{4} + \frac{- 1 \cdot (x \sqrt{7}) \cdot 4 - \sqrt{7}}{4}) (x + 6 i) (x - 6 i)$

Step 7.2.5

Combine the numerators over the common denominator.

$y = (\frac{x}{4} + \frac{x^{2} \cdot 4 - 1 \cdot (x \sqrt{7}) \cdot 4 - \sqrt{7}}{4}) (x + 6 i) (x - 6 i)$

Step 7.2.6

Combine the numerators over the common denominator.

$y = \frac{x + x^{2} \cdot 4 - 1 \cdot (x \sqrt{7}) \cdot 4 - \sqrt{7}}{4} \cdot ((x + 6 i) (x - 6 i))$

Step 7.3

Simplify the numerator.

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Step 7.3.1

Reorder terms.

$y = \frac{4 x^{2} - 1 \cdot (4 x \sqrt{7}) + x - \sqrt{7}}{4} \cdot ((x + 6 i) (x - 6 i))$

Step 7.3.2

Factor out the greatest common factor from each group.

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Step 7.3.2.1

Group the first two terms and the last two terms.

$y = \frac{(4 x^{2} - 1 \cdot (4 x \sqrt{7})) + x - \sqrt{7}}{4} \cdot ((x + 6 i) (x - 6 i))$

Step 7.3.2.2

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

$y = \frac{4 x (x - \sqrt{7}) + 1 (x - \sqrt{7})}{4} \cdot ((x + 6 i) (x - 6 i))$

Step 7.3.3

Factor the polynomial by factoring out the greatest common factor, $x - \sqrt{7}$ .

$y = \frac{(x - \sqrt{7}) (4 x + 1)}{4} \cdot ((x + 6 i) (x - 6 i))$

Step 7.4

Simplify terms.

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Step 7.4.1

Apply the distributive property.

$y = (\frac{(x - \sqrt{7}) (4 x + 1)}{4} \cdot x + \frac{(x - \sqrt{7}) (4 x + 1)}{4} \cdot (6 i)) (x - 6 i)$

Step 7.4.2

Combine $\frac{(x - \sqrt{7}) (4 x + 1)}{4}$ and $x$ .

$y = (\frac{(x - \sqrt{7}) (4 x + 1) x}{4} + \frac{(x - \sqrt{7}) (4 x + 1)}{4} \cdot (6 i)) (x - 6 i)$

Step 7.4.3

Cancel the common factor of $2$ .

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Step 7.4.3.1

Factor $2$ out of $4$ .

$y = (\frac{(x - \sqrt{7}) (4 x + 1) x}{4} + \frac{(x - \sqrt{7}) (4 x + 1)}{2 (2)} \cdot (6 i)) (x - 6 i)$

Step 7.4.3.2

Factor $2$ out of $6 i$ .

$y = (\frac{(x - \sqrt{7}) (4 x + 1) x}{4} + \frac{(x - \sqrt{7}) (4 x + 1)}{2 (2)} \cdot (2 (3 i))) (x - 6 i)$

Step 7.4.3.3

Cancel the common factor.

$y = (\frac{(x - \sqrt{7}) (4 x + 1) x}{4} + \frac{(x - \sqrt{7}) (4 x + 1)}{2 \cdot 2} \cdot (2 (3 i))) (x - 6 i)$

Step 7.4.3.4

Rewrite the expression.

$y = (\frac{(x - \sqrt{7}) (4 x + 1) x}{4} + \frac{(x - \sqrt{7}) (4 x + 1)}{2} \cdot (3 i)) (x - 6 i)$

Step 7.4.4

Combine $3$ and $\frac{(x - \sqrt{7}) (4 x + 1)}{2}$ .

$y = (\frac{(x - \sqrt{7}) (4 x + 1) x}{4} + \frac{3 ((x - \sqrt{7}) (4 x + 1))}{2} \cdot i) (x - 6 i)$

Step 7.4.5

Combine $\frac{3 ((x - \sqrt{7}) (4 x + 1))}{2}$ and $i$ .

$y = (\frac{(x - \sqrt{7}) (4 x + 1) x}{4} + \frac{3 ((x - \sqrt{7}) (4 x + 1)) i}{2}) (x - 6 i)$

Step 7.5

To write $\frac{3 (x - \sqrt{7}) (4 x + 1) i}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = (\frac{(x - \sqrt{7}) (4 x + 1) x}{4} + \frac{3 (x - \sqrt{7}) (4 x + 1) i}{2} \cdot \frac{2}{2}) (x - 6 i)$

Step 7.6

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.6.1

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

$y = (\frac{(x - \sqrt{7}) (4 x + 1) x}{4} + \frac{3 (x - \sqrt{7}) (4 x + 1) i \cdot 2}{2 \cdot 2}) (x - 6 i)$

Step 7.6.2

Multiply $2$ by $2$ .

$y = (\frac{(x - \sqrt{7}) (4 x + 1) x}{4} + \frac{3 (x - \sqrt{7}) (4 x + 1) i \cdot 2}{4}) (x - 6 i)$

Step 7.7

Combine the numerators over the common denominator.

$y = \frac{(x - \sqrt{7}) (4 x + 1) x + 3 (x - \sqrt{7}) (4 x + 1) i \cdot 2}{4} \cdot (x - 6 i)$

Step 7.8

Simplify the numerator.

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Step 7.8.1

Factor $(x - \sqrt{7}) (4 x + 1)$ out of $(x - \sqrt{7}) (4 x + 1) x + 3 (x - \sqrt{7}) (4 x + 1) i \cdot 2$ .

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Step 7.8.1.1

Factor $(x - \sqrt{7}) (4 x + 1)$ out of $(x - \sqrt{7}) (4 x + 1) x$ .

$y = \frac{(x - \sqrt{7}) (4 x + 1) (x) + 3 (x - \sqrt{7}) (4 x + 1) i \cdot 2}{4} \cdot (x - 6 i)$

Step 7.8.1.2

Factor $(x - \sqrt{7}) (4 x + 1)$ out of $3 (x - \sqrt{7}) (4 x + 1) i \cdot 2$ .

$y = \frac{(x - \sqrt{7}) (4 x + 1) (x) + (x - \sqrt{7}) (4 x + 1) ((3) i \cdot 2)}{4} \cdot (x - 6 i)$

Step 7.8.1.3

Factor $(x - \sqrt{7}) (4 x + 1)$ out of $(x - \sqrt{7}) (4 x + 1) (x) + (x - \sqrt{7}) (4 x + 1) ((3) i \cdot 2)$ .

$y = \frac{(x - \sqrt{7}) (4 x + 1) (x + (3) i \cdot 2)}{4} \cdot (x - 6 i)$

Step 7.8.2

Multiply $2$ by $3$ .

$y = \frac{(x - \sqrt{7}) (4 x + 1) (x + 6 i)}{4} \cdot (x - 6 i)$

Step 7.9

Simplify terms.

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Step 7.9.1

Apply the distributive property.

$y = \frac{(x - \sqrt{7}) (4 x + 1) (x + 6 i)}{4} \cdot x + \frac{(x - \sqrt{7}) (4 x + 1) (x + 6 i)}{4} \cdot (- 6 i)$

Step 7.9.2

Simplify terms.

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Step 7.9.2.1

Combine $\frac{(x - \sqrt{7}) (4 x + 1) (x + 6 i)}{4}$ and $x$ .

$y = \frac{(x - \sqrt{7}) (4 x + 1) (x + 6 i) x}{4} + \frac{(x - \sqrt{7}) (4 x + 1) (x + 6 i)}{4} \cdot (- 6 i)$

Step 7.9.2.2

Cancel the common factor of $2$ .

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Step 7.9.2.2.1

Factor $2$ out of $4$ .

$y = \frac{(x - \sqrt{7}) (4 x + 1) (x + 6 i) x}{4} + \frac{(x - \sqrt{7}) (4 x + 1) (x + 6 i)}{2 (2)} \cdot (- 6 i)$

Step 7.9.2.2.2

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

$y = \frac{(x - \sqrt{7}) (4 x + 1) (x + 6 i) x}{4} + \frac{(x - \sqrt{7}) (4 x + 1) (x + 6 i)}{2 (2)} \cdot (2 (- 3 i))$

Step 7.9.2.2.3

Cancel the common factor.

$y = \frac{(x - \sqrt{7}) (4 x + 1) (x + 6 i) x}{4} + \frac{(x - \sqrt{7}) (4 x + 1) (x + 6 i)}{2 \cdot 2} \cdot (2 (- 3 i))$

Step 7.9.2.2.4

Rewrite the expression.

$y = \frac{(x - \sqrt{7}) (4 x + 1) (x + 6 i) x}{4} + \frac{(x - \sqrt{7}) (4 x + 1) (x + 6 i)}{2} \cdot (- 3 i)$

Step 7.9.2.3

Combine $- 3$ and $\frac{(x - \sqrt{7}) (4 x + 1) (x + 6 i)}{2}$ .

$y = \frac{(x - \sqrt{7}) (4 x + 1) (x + 6 i) x}{4} + \frac{- 3 ((x - \sqrt{7}) (4 x + 1) (x + 6 i))}{2} \cdot i$

Step 7.9.2.4

Combine $\frac{- 3 ((x - \sqrt{7}) (4 x + 1) (x + 6 i))}{2}$ and $i$ .

$y = \frac{(x - \sqrt{7}) (4 x + 1) (x + 6 i) x}{4} + \frac{- 3 ((x - \sqrt{7}) (4 x + 1) (x + 6 i)) i}{2}$

Step 7.9.3

Move the negative in front of the fraction.

$y = \frac{(x - \sqrt{7}) (4 x + 1) (x + 6 i) x}{4} - \frac{3 (x - \sqrt{7}) (4 x + 1) (x + 6 i) i}{2}$

Step 7.10

To write $- \frac{3 (x - \sqrt{7}) (4 x + 1) (x + 6 i) i}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{(x - \sqrt{7}) (4 x + 1) (x + 6 i) x}{4} - \frac{3 (x - \sqrt{7}) (4 x + 1) (x + 6 i) i}{2} \cdot \frac{2}{2}$

Step 7.11

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.11.1

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

$y = \frac{(x - \sqrt{7}) (4 x + 1) (x + 6 i) x}{4} - \frac{3 (x - \sqrt{7}) (4 x + 1) (x + 6 i) i \cdot 2}{2 \cdot 2}$

Step 7.11.2

Multiply $2$ by $2$ .

$y = \frac{(x - \sqrt{7}) (4 x + 1) (x + 6 i) x}{4} - \frac{3 (x - \sqrt{7}) (4 x + 1) (x + 6 i) i \cdot 2}{4}$

Step 7.12

Combine the numerators over the common denominator.

$y = \frac{(x - \sqrt{7}) (4 x + 1) (x + 6 i) x - 3 (x - \sqrt{7}) (4 x + 1) (x + 6 i) i \cdot 2}{4}$

Step 7.13

Multiply $2$ by $- 3$ .

$y = \frac{(x - \sqrt{7}) (4 x + 1) (x + 6 i) x - 6 (x - \sqrt{7}) (4 x + 1) (x + 6 i) i}{4}$

Step 7.14

Simplify each term.

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Step 7.14.1

Expand $(x - \sqrt{7}) (4 x + 1)$ using the FOIL Method.

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Step 7.14.1.1

Apply the distributive property.

$y = \frac{(x (4 x + 1) - \sqrt{7} (4 x + 1)) (x + 6 i) x - 6 (x - \sqrt{7}) (4 x + 1) (x + 6 i) i}{4}$

Step 7.14.1.2

Apply the distributive property.

$y = \frac{(x (4 x) + x \cdot 1 - \sqrt{7} (4 x + 1)) (x + 6 i) x - 6 (x - \sqrt{7}) (4 x + 1) (x + 6 i) i}{4}$

Step 7.14.1.3

Apply the distributive property.

$y = \frac{(x (4 x) + x \cdot 1 - \sqrt{7} (4 x) - \sqrt{7} \cdot 1) (x + 6 i) x - 6 (x - \sqrt{7}) (4 x + 1) (x + 6 i) i}{4}$

Step 7.14.2

Simplify each term.

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Step 7.14.2.1

Rewrite using the commutative property of multiplication.

$y = \frac{(4 x \cdot x + x \cdot 1 - \sqrt{7} (4 x) - \sqrt{7} \cdot 1) (x + 6 i) x - 6 (x - \sqrt{7}) (4 x + 1) (x + 6 i) i}{4}$

Step 7.14.2.2

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

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Step 7.14.2.2.1

Move $x$ .

$y = \frac{(4 (x \cdot x) + x \cdot 1 - \sqrt{7} (4 x) - \sqrt{7} \cdot 1) (x + 6 i) x - 6 (x - \sqrt{7}) (4 x + 1) (x + 6 i) i}{4}$

Step 7.14.2.2.2

Multiply $x$ by $x$ .

$y = \frac{(4 x^{2} + x \cdot 1 - \sqrt{7} (4 x) - \sqrt{7} \cdot 1) (x + 6 i) x - 6 (x - \sqrt{7}) (4 x + 1) (x + 6 i) i}{4}$

Step 7.14.2.3

Multiply $x$ by $1$ .

$y = \frac{(4 x^{2} + x - \sqrt{7} (4 x) - \sqrt{7} \cdot 1) (x + 6 i) x - 6 (x - \sqrt{7}) (4 x + 1) (x + 6 i) i}{4}$

Step 7.14.2.4

Multiply $4$ by $- 1$ .

$y = \frac{(4 x^{2} + x - 4 \sqrt{7} x - \sqrt{7} \cdot 1) (x + 6 i) x - 6 (x - \sqrt{7}) (4 x + 1) (x + 6 i) i}{4}$

Step 7.14.2.5

Multiply $- 1$ by $1$ .

$y = \frac{(4 x^{2} + x - 4 \sqrt{7} x - \sqrt{7}) (x + 6 i) x - 6 (x - \sqrt{7}) (4 x + 1) (x + 6 i) i}{4}$

Step 7.14.3

Expand $(4 x^{2} + x - 4 \sqrt{7} x - \sqrt{7}) (x + 6 i)$ by multiplying each term in the first expression by each term in the second expression.

$y = \frac{(4 x^{2} x + 4 x^{2} (6 i) + x \cdot x + x (6 i) - 4 \sqrt{7} x \cdot x - 4 \sqrt{7} x (6 i) - \sqrt{7} x - \sqrt{7} (6 i)) x - 6 (x - \sqrt{7}) (4 x + 1) (x + 6 i) i}{4}$

Step 7.14.4

Simplify each term.

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Step 7.14.4.1

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

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Step 7.14.4.1.1

Move $x$ .

$y = \frac{(4 (x \cdot x^{2}) + 4 x^{2} (6 i) + x \cdot x + x (6 i) - 4 \sqrt{7} x \cdot x - 4 \sqrt{7} x (6 i) - \sqrt{7} x - \sqrt{7} (6 i)) x - 6 (x - \sqrt{7}) (4 x + 1) (x + 6 i) i}{4}$

Step 7.14.4.1.2

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

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Step 7.14.4.1.2.1

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

$y = \frac{(4 (x \cdot x^{2}) + 4 x^{2} (6 i) + x \cdot x + x (6 i) - 4 \sqrt{7} x \cdot x - 4 \sqrt{7} x (6 i) - \sqrt{7} x - \sqrt{7} (6 i)) x - 6 (x - \sqrt{7}) (4 x + 1) (x + 6 i) i}{4}$

Step 7.14.4.1.2.2

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

$y = \frac{(4 x^{1 + 2} + 4 x^{2} (6 i) + x \cdot x + x (6 i) - 4 \sqrt{7} x \cdot x - 4 \sqrt{7} x (6 i) - \sqrt{7} x - \sqrt{7} (6 i)) x - 6 (x - \sqrt{7}) (4 x + 1) (x + 6 i) i}{4}$

Step 7.14.4.1.3

Add $1$ and $2$ .

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

Step 7.14.4.2

Multiply $6$ by $4$ .

$y = \frac{(4 x^{3} + 24 x^{2} i + x \cdot x + x (6 i) - 4 \sqrt{7} x \cdot x - 4 \sqrt{7} x (6 i) - \sqrt{7} x - \sqrt{7} (6 i)) x - 6 (x - \sqrt{7}) (4 x + 1) (x + 6 i) i}{4}$

Step 7.14.4.3

Multiply $x$ by $x$ .

$y = \frac{(4 x^{3} + 24 x^{2} i + x^{2} + x (6 i) - 4 \sqrt{7} x \cdot x - 4 \sqrt{7} x (6 i) - \sqrt{7} x - \sqrt{7} (6 i)) x - 6 (x - \sqrt{7}) (4 x + 1) (x + 6 i) i}{4}$

Step 7.14.4.4

Move $6$ to the left of $x$ .

$y = \frac{(4 x^{3} + 24 x^{2} i + x^{2} + 6 \cdot (x i) - 4 \sqrt{7} x \cdot x - 4 \sqrt{7} x (6 i) - \sqrt{7} x - \sqrt{7} (6 i)) x - 6 (x - \sqrt{7}) (4 x + 1) (x + 6 i) i}{4}$

Step 7.14.4.5

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

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Step 7.14.4.5.1

Move $x$ .

$y = \frac{(4 x^{3} + 24 x^{2} i + x^{2} + 6 x i - 4 \sqrt{7} (x \cdot x) - 4 \sqrt{7} x (6 i) - \sqrt{7} x - \sqrt{7} (6 i)) x - 6 (x - \sqrt{7}) (4 x + 1) (x + 6 i) i}{4}$

Step 7.14.4.5.2

Multiply $x$ by $x$ .

$y = \frac{(4 x^{3} + 24 x^{2} i + x^{2} + 6 x i - 4 \sqrt{7} x^{2} - 4 \sqrt{7} x (6 i) - \sqrt{7} x - \sqrt{7} (6 i)) x - 6 (x - \sqrt{7}) (4 x + 1) (x + 6 i) i}{4}$

Step 7.14.4.6

Multiply $6$ by $- 4$ .

$y = \frac{(4 x^{3} + 24 x^{2} i + x^{2} + 6 x i - 4 \sqrt{7} x^{2} - 24 \sqrt{7} x i - \sqrt{7} x - \sqrt{7} (6 i)) x - 6 (x - \sqrt{7}) (4 x + 1) (x + 6 i) i}{4}$

Step 7.14.4.7

Multiply $6$ by $- 1$ .

$y = \frac{(4 x^{3} + 24 x^{2} i + x^{2} + 6 x i - 4 \sqrt{7} x^{2} - 24 \sqrt{7} x i - \sqrt{7} x - 6 \sqrt{7} i) x - 6 (x - \sqrt{7}) (4 x + 1) (x + 6 i) i}{4}$

Step 7.14.5

Apply the distributive property.

$y = \frac{4 x^{3} x + 24 x^{2} i x + x^{2} x + 6 x i x - 4 \sqrt{7} x^{2} x - 24 \sqrt{7} x i x - \sqrt{7} x \cdot x - 6 \sqrt{7} i x - 6 (x - \sqrt{7}) (4 x + 1) (x + 6 i) i}{4}$

Step 7.14.6

Simplify.

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Step 7.14.6.1

Multiply $x^{3}$ by $x$ by adding the exponents.

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Step 7.14.6.1.1

Move $x$ .

$y = \frac{4 (x \cdot x^{3}) + 24 x^{2} i x + x^{2} x + 6 x i x - 4 \sqrt{7} x^{2} x - 24 \sqrt{7} x i x - \sqrt{7} x \cdot x - 6 \sqrt{7} i x - 6 (x - \sqrt{7}) (4 x + 1) (x + 6 i) i}{4}$

Step 7.14.6.1.2

Multiply $x$ by $x^{3}$ .

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Step 7.14.6.1.2.1

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

$y = \frac{4 (x \cdot x^{3}) + 24 x^{2} i x + x^{2} x + 6 x i x - 4 \sqrt{7} x^{2} x - 24 \sqrt{7} x i x - \sqrt{7} x \cdot x - 6 \sqrt{7} i x - 6 (x - \sqrt{7}) (4 x + 1) (x + 6 i) i}{4}$

Step 7.14.6.1.2.2

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

$y = \frac{4 x^{1 + 3} + 24 x^{2} i x + x^{2} x + 6 x i x - 4 \sqrt{7} x^{2} x - 24 \sqrt{7} x i x - \sqrt{7} x \cdot x - 6 \sqrt{7} i x - 6 (x - \sqrt{7}) (4 x + 1) (x + 6 i) i}{4}$

Step 7.14.6.1.3

Add $1$ and $3$ .

$y = \frac{4 x^{4} + 24 x^{2} i x + x^{2} x + 6 x i x - 4 \sqrt{7} x^{2} x - 24 \sqrt{7} x i x - \sqrt{7} x \cdot x - 6 \sqrt{7} i x - 6 (x - \sqrt{7}) (4 x + 1) (x + 6 i) i}{4}$

Step 7.14.6.2

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

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Step 7.14.6.2.1

Move $x$ .

$y = \frac{4 x^{4} + 24 (x \cdot x^{2}) i + x^{2} x + 6 x i x - 4 \sqrt{7} x^{2} x - 24 \sqrt{7} x i x - \sqrt{7} x \cdot x - 6 \sqrt{7} i x - 6 (x - \sqrt{7}) (4 x + 1) (x + 6 i) i}{4}$

Step 7.14.6.2.2

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

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Step 7.14.6.2.2.1

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

$y = \frac{4 x^{4} + 24 (x \cdot x^{2}) i + x^{2} x + 6 x i x - 4 \sqrt{7} x^{2} x - 24 \sqrt{7} x i x - \sqrt{7} x \cdot x - 6 \sqrt{7} i x - 6 (x - \sqrt{7}) (4 x + 1) (x + 6 i) i}{4}$

Step 7.14.6.2.2.2

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

$y = \frac{4 x^{4} + 24 x^{1 + 2} i + x^{2} x + 6 x i x - 4 \sqrt{7} x^{2} x - 24 \sqrt{7} x i x - \sqrt{7} x \cdot x - 6 \sqrt{7} i x - 6 (x - \sqrt{7}) (4 x + 1) (x + 6 i) i}{4}$

Step 7.14.6.2.3

Add $1$ and $2$ .

$y = \frac{4 x^{4} + 24 x^{3} i + x^{2} x + 6 x i x - 4 \sqrt{7} x^{2} x - 24 \sqrt{7} x i x - \sqrt{7} x \cdot x - 6 \sqrt{7} i x - 6 (x - \sqrt{7}) (4 x + 1) (x + 6 i) i}{4}$

Step 7.14.6.3

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

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Step 7.14.6.3.1

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

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Step 7.14.6.3.1.1

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

$y = \frac{4 x^{4} + 24 x^{3} i + x^{2} x + 6 x i x - 4 \sqrt{7} x^{2} x - 24 \sqrt{7} x i x - \sqrt{7} x \cdot x - 6 \sqrt{7} i x - 6 (x - \sqrt{7}) (4 x + 1) (x + 6 i) i}{4}$

Step 7.14.6.3.1.2

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

$y = \frac{4 x^{4} + 24 x^{3} i + x^{2 + 1} + 6 x i x - 4 \sqrt{7} x^{2} x - 24 \sqrt{7} x i x - \sqrt{7} x \cdot x - 6 \sqrt{7} i x - 6 (x - \sqrt{7}) (4 x + 1) (x + 6 i) i}{4}$

Step 7.14.6.3.2

Add $2$ and $1$ .

$y = \frac{4 x^{4} + 24 x^{3} i + x^{3} + 6 x i x - 4 \sqrt{7} x^{2} x - 24 \sqrt{7} x i x - \sqrt{7} x \cdot x - 6 \sqrt{7} i x - 6 (x - \sqrt{7}) (4 x + 1) (x + 6 i) i}{4}$

Step 7.14.6.4

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

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Step 7.14.6.4.1

Move $x$ .

$y = \frac{4 x^{4} + 24 x^{3} i + x^{3} + 6 (x \cdot x) i - 4 \sqrt{7} x^{2} x - 24 \sqrt{7} x i x - \sqrt{7} x \cdot x - 6 \sqrt{7} i x - 6 (x - \sqrt{7}) (4 x + 1) (x + 6 i) i}{4}$

Step 7.14.6.4.2

Multiply $x$ by $x$ .

$y = \frac{4 x^{4} + 24 x^{3} i + x^{3} + 6 x^{2} i - 4 \sqrt{7} x^{2} x - 24 \sqrt{7} x i x - \sqrt{7} x \cdot x - 6 \sqrt{7} i x - 6 (x - \sqrt{7}) (4 x + 1) (x + 6 i) i}{4}$

Step 7.14.6.5

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

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Step 7.14.6.5.1

Move $x$ .

$y = \frac{4 x^{4} + 24 x^{3} i + x^{3} + 6 x^{2} i - 4 \sqrt{7} (x \cdot x^{2}) - 24 \sqrt{7} x i x - \sqrt{7} x \cdot x - 6 \sqrt{7} i x - 6 (x - \sqrt{7}) (4 x + 1) (x + 6 i) i}{4}$

Step 7.14.6.5.2

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

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Step 7.14.6.5.2.1

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

$y = \frac{4 x^{4} + 24 x^{3} i + x^{3} + 6 x^{2} i - 4 \sqrt{7} (x \cdot x^{2}) - 24 \sqrt{7} x i x - \sqrt{7} x \cdot x - 6 \sqrt{7} i x - 6 (x - \sqrt{7}) (4 x + 1) (x + 6 i) i}{4}$

Step 7.14.6.5.2.2

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

$y = \frac{4 x^{4} + 24 x^{3} i + x^{3} + 6 x^{2} i - 4 \sqrt{7} x^{1 + 2} - 24 \sqrt{7} x i x - \sqrt{7} x \cdot x - 6 \sqrt{7} i x - 6 (x - \sqrt{7}) (4 x + 1) (x + 6 i) i}{4}$

Step 7.14.6.5.3

Add $1$ and $2$ .

$y = \frac{4 x^{4} + 24 x^{3} i + x^{3} + 6 x^{2} i - 4 \sqrt{7} x^{3} - 24 \sqrt{7} x i x - \sqrt{7} x \cdot x - 6 \sqrt{7} i x - 6 (x - \sqrt{7}) (4 x + 1) (x + 6 i) i}{4}$

Step 7.14.6.6

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

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Step 7.14.6.6.1

Move $x$ .

$y = \frac{4 x^{4} + 24 x^{3} i + x^{3} + 6 x^{2} i - 4 \sqrt{7} x^{3} - 24 \sqrt{7} (x \cdot x) i - \sqrt{7} x \cdot x - 6 \sqrt{7} i x - 6 (x - \sqrt{7}) (4 x + 1) (x + 6 i) i}{4}$

Step 7.14.6.6.2

Multiply $x$ by $x$ .

$y = \frac{4 x^{4} + 24 x^{3} i + x^{3} + 6 x^{2} i - 4 \sqrt{7} x^{3} - 24 \sqrt{7} x^{2} i - \sqrt{7} x \cdot x - 6 \sqrt{7} i x - 6 (x - \sqrt{7}) (4 x + 1) (x + 6 i) i}{4}$

Step 7.14.6.7

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

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Step 7.14.6.7.1

Move $x$ .

$y = \frac{4 x^{4} + 24 x^{3} i + x^{3} + 6 x^{2} i - 4 \sqrt{7} x^{3} - 24 \sqrt{7} x^{2} i - \sqrt{7} (x \cdot x) - 6 \sqrt{7} i x - 6 (x - \sqrt{7}) (4 x + 1) (x + 6 i) i}{4}$

Step 7.14.6.7.2

Multiply $x$ by $x$ .

$y = \frac{4 x^{4} + 24 x^{3} i + x^{3} + 6 x^{2} i - 4 \sqrt{7} x^{3} - 24 \sqrt{7} x^{2} i - \sqrt{7} x^{2} - 6 \sqrt{7} i x - 6 (x - \sqrt{7}) (4 x + 1) (x + 6 i) i}{4}$

Step 7.14.7

Apply the distributive property.

$y = \frac{4 x^{4} + 24 x^{3} i + x^{3} + 6 x^{2} i - 4 \sqrt{7} x^{3} - 24 \sqrt{7} x^{2} i - \sqrt{7} x^{2} - 6 \sqrt{7} i x + (- 6 x - 6 (- \sqrt{7})) (4 x + 1) (x + 6 i) i}{4}$

Step 7.14.8

Multiply $- 1$ by $- 6$ .

$y = \frac{4 x^{4} + 24 x^{3} i + x^{3} + 6 x^{2} i - 4 \sqrt{7} x^{3} - 24 \sqrt{7} x^{2} i - \sqrt{7} x^{2} - 6 \sqrt{7} i x + (- 6 x + 6 \sqrt{7}) (4 x + 1) (x + 6 i) i}{4}$

Step 7.14.9

Expand $(- 6 x + 6 \sqrt{7}) (4 x + 1)$ using the FOIL Method.

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Step 7.14.9.1

Apply the distributive property.

$y = \frac{4 x^{4} + 24 x^{3} i + x^{3} + 6 x^{2} i - 4 \sqrt{7} x^{3} - 24 \sqrt{7} x^{2} i - \sqrt{7} x^{2} - 6 \sqrt{7} i x + (- 6 x (4 x + 1) + 6 \sqrt{7} (4 x + 1)) (x + 6 i) i}{4}$

Step 7.14.9.2

Apply the distributive property.

$y = \frac{4 x^{4} + 24 x^{3} i + x^{3} + 6 x^{2} i - 4 \sqrt{7} x^{3} - 24 \sqrt{7} x^{2} i - \sqrt{7} x^{2} - 6 \sqrt{7} i x + (- 6 x (4 x) - 6 x \cdot 1 + 6 \sqrt{7} (4 x + 1)) (x + 6 i) i}{4}$

Step 7.14.9.3

Apply the distributive property.

$y = \frac{4 x^{4} + 24 x^{3} i + x^{3} + 6 x^{2} i - 4 \sqrt{7} x^{3} - 24 \sqrt{7} x^{2} i - \sqrt{7} x^{2} - 6 \sqrt{7} i x + (- 6 x (4 x) - 6 x \cdot 1 + 6 \sqrt{7} (4 x) + 6 \sqrt{7} \cdot 1) (x + 6 i) i}{4}$

Step 7.14.10

Simplify each term.

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Step 7.14.10.1

Rewrite using the commutative property of multiplication.

$y = \frac{4 x^{4} + 24 x^{3} i + x^{3} + 6 x^{2} i - 4 \sqrt{7} x^{3} - 24 \sqrt{7} x^{2} i - \sqrt{7} x^{2} - 6 \sqrt{7} i x + (- 6 \cdot (4 x \cdot x) - 6 x \cdot 1 + 6 \sqrt{7} (4 x) + 6 \sqrt{7} \cdot 1) (x + 6 i) i}{4}$

Step 7.14.10.2

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

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Step 7.14.10.2.1

Move $x$ .

$y = \frac{4 x^{4} + 24 x^{3} i + x^{3} + 6 x^{2} i - 4 \sqrt{7} x^{3} - 24 \sqrt{7} x^{2} i - \sqrt{7} x^{2} - 6 \sqrt{7} i x + (- 6 \cdot (4 (x \cdot x)) - 6 x \cdot 1 + 6 \sqrt{7} (4 x) + 6 \sqrt{7} \cdot 1) (x + 6 i) i}{4}$

Step 7.14.10.2.2

Multiply $x$ by $x$ .

$y = \frac{4 x^{4} + 24 x^{3} i + x^{3} + 6 x^{2} i - 4 \sqrt{7} x^{3} - 24 \sqrt{7} x^{2} i - \sqrt{7} x^{2} - 6 \sqrt{7} i x + (- 6 \cdot (4 x^{2}) - 6 x \cdot 1 + 6 \sqrt{7} (4 x) + 6 \sqrt{7} \cdot 1) (x + 6 i) i}{4}$

Step 7.14.10.3

Multiply $- 6$ by $4$ .

$y = \frac{4 x^{4} + 24 x^{3} i + x^{3} + 6 x^{2} i - 4 \sqrt{7} x^{3} - 24 \sqrt{7} x^{2} i - \sqrt{7} x^{2} - 6 \sqrt{7} i x + (- 24 x^{2} - 6 x \cdot 1 + 6 \sqrt{7} (4 x) + 6 \sqrt{7} \cdot 1) (x + 6 i) i}{4}$

Step 7.14.10.4

Multiply $- 6$ by $1$ .

$y = \frac{4 x^{4} + 24 x^{3} i + x^{3} + 6 x^{2} i - 4 \sqrt{7} x^{3} - 24 \sqrt{7} x^{2} i - \sqrt{7} x^{2} - 6 \sqrt{7} i x + (- 24 x^{2} - 6 x + 6 \sqrt{7} (4 x) + 6 \sqrt{7} \cdot 1) (x + 6 i) i}{4}$

Step 7.14.10.5

Multiply $4$ by $6$ .

$y = \frac{4 x^{4} + 24 x^{3} i + x^{3} + 6 x^{2} i - 4 \sqrt{7} x^{3} - 24 \sqrt{7} x^{2} i - \sqrt{7} x^{2} - 6 \sqrt{7} i x + (- 24 x^{2} - 6 x + 24 \sqrt{7} x + 6 \sqrt{7} \cdot 1) (x + 6 i) i}{4}$

Step 7.14.10.6

Multiply $6$ by $1$ .

$y = \frac{4 x^{4} + 24 x^{3} i + x^{3} + 6 x^{2} i - 4 \sqrt{7} x^{3} - 24 \sqrt{7} x^{2} i - \sqrt{7} x^{2} - 6 \sqrt{7} i x + (- 24 x^{2} - 6 x + 24 \sqrt{7} x + 6 \sqrt{7}) (x + 6 i) i}{4}$

Step 7.14.11

Expand $(- 24 x^{2} - 6 x + 24 \sqrt{7} x + 6 \sqrt{7}) (x + 6 i)$ by multiplying each term in the first expression by each term in the second expression.

$y = \frac{4 x^{4} + 24 x^{3} i + x^{3} + 6 x^{2} i - 4 \sqrt{7} x^{3} - 24 \sqrt{7} x^{2} i - \sqrt{7} x^{2} - 6 \sqrt{7} i x + (- 24 x^{2} x - 24 x^{2} (6 i) - 6 x \cdot x - 6 x (6 i) + 24 \sqrt{7} x \cdot x + 24 \sqrt{7} x (6 i) + 6 \sqrt{7} x + 6 \sqrt{7} (6 i)) i}{4}$

Step 7.14.12

Simplify each term.

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Step 7.14.12.1

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

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Step 7.14.12.1.1

Move $x$ .

$y = \frac{4 x^{4} + 24 x^{3} i + x^{3} + 6 x^{2} i - 4 \sqrt{7} x^{3} - 24 \sqrt{7} x^{2} i - \sqrt{7} x^{2} - 6 \sqrt{7} i x + (- 24 (x \cdot x^{2}) - 24 x^{2} (6 i) - 6 x \cdot x - 6 x (6 i) + 24 \sqrt{7} x \cdot x + 24 \sqrt{7} x (6 i) + 6 \sqrt{7} x + 6 \sqrt{7} (6 i)) i}{4}$

Step 7.14.12.1.2

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

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Step 7.14.12.1.2.1

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

Step 7.14.12.1.2.2

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

$y = \frac{4 x^{4} + 24 x^{3} i + x^{3} + 6 x^{2} i - 4 \sqrt{7} x^{3} - 24 \sqrt{7} x^{2} i - \sqrt{7} x^{2} - 6 \sqrt{7} i x + (- 24 x^{1 + 2} - 24 x^{2} (6 i) - 6 x \cdot x - 6 x (6 i) + 24 \sqrt{7} x \cdot x + 24 \sqrt{7} x (6 i) + 6 \sqrt{7} x + 6 \sqrt{7} (6 i)) i}{4}$

Step 7.14.12.1.3

Add $1$ and $2$ .

$y = \frac{4 x^{4} + 24 x^{3} i + x^{3} + 6 x^{2} i - 4 \sqrt{7} x^{3} - 24 \sqrt{7} x^{2} i - \sqrt{7} x^{2} - 6 \sqrt{7} i x + (- 24 x^{3} - 24 x^{2} (6 i) - 6 x \cdot x - 6 x (6 i) + 24 \sqrt{7} x \cdot x + 24 \sqrt{7} x (6 i) + 6 \sqrt{7} x + 6 \sqrt{7} (6 i)) i}{4}$

Step 7.14.12.2

Multiply $6$ by $- 24$ .

$y = \frac{4 x^{4} + 24 x^{3} i + x^{3} + 6 x^{2} i - 4 \sqrt{7} x^{3} - 24 \sqrt{7} x^{2} i - \sqrt{7} x^{2} - 6 \sqrt{7} i x + (- 24 x^{3} - 144 x^{2} i - 6 x \cdot x - 6 x (6 i) + 24 \sqrt{7} x \cdot x + 24 \sqrt{7} x (6 i) + 6 \sqrt{7} x + 6 \sqrt{7} (6 i)) i}{4}$

Step 7.14.12.3

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

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Step 7.14.12.3.1

Move $x$ .

$y = \frac{4 x^{4} + 24 x^{3} i + x^{3} + 6 x^{2} i - 4 \sqrt{7} x^{3} - 24 \sqrt{7} x^{2} i - \sqrt{7} x^{2} - 6 \sqrt{7} i x + (- 24 x^{3} - 144 x^{2} i - 6 (x \cdot x) - 6 x (6 i) + 24 \sqrt{7} x \cdot x + 24 \sqrt{7} x (6 i) + 6 \sqrt{7} x + 6 \sqrt{7} (6 i)) i}{4}$

Step 7.14.12.3.2

Multiply $x$ by $x$ .

$y = \frac{4 x^{4} + 24 x^{3} i + x^{3} + 6 x^{2} i - 4 \sqrt{7} x^{3} - 24 \sqrt{7} x^{2} i - \sqrt{7} x^{2} - 6 \sqrt{7} i x + (- 24 x^{3} - 144 x^{2} i - 6 x^{2} - 6 x (6 i) + 24 \sqrt{7} x \cdot x + 24 \sqrt{7} x (6 i) + 6 \sqrt{7} x + 6 \sqrt{7} (6 i)) i}{4}$

Step 7.14.12.4

Multiply $6$ by $- 6$ .

$y = \frac{4 x^{4} + 24 x^{3} i + x^{3} + 6 x^{2} i - 4 \sqrt{7} x^{3} - 24 \sqrt{7} x^{2} i - \sqrt{7} x^{2} - 6 \sqrt{7} i x + (- 24 x^{3} - 144 x^{2} i - 6 x^{2} - 36 x i + 24 \sqrt{7} x \cdot x + 24 \sqrt{7} x (6 i) + 6 \sqrt{7} x + 6 \sqrt{7} (6 i)) i}{4}$

Step 7.14.12.5

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

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Step 7.14.12.5.1

Move $x$ .

$y = \frac{4 x^{4} + 24 x^{3} i + x^{3} + 6 x^{2} i - 4 \sqrt{7} x^{3} - 24 \sqrt{7} x^{2} i - \sqrt{7} x^{2} - 6 \sqrt{7} i x + (- 24 x^{3} - 144 x^{2} i - 6 x^{2} - 36 x i + 24 \sqrt{7} (x \cdot x) + 24 \sqrt{7} x (6 i) + 6 \sqrt{7} x + 6 \sqrt{7} (6 i)) i}{4}$

Step 7.14.12.5.2

Multiply $x$ by $x$ .

$y = \frac{4 x^{4} + 24 x^{3} i + x^{3} + 6 x^{2} i - 4 \sqrt{7} x^{3} - 24 \sqrt{7} x^{2} i - \sqrt{7} x^{2} - 6 \sqrt{7} i x + (- 24 x^{3} - 144 x^{2} i - 6 x^{2} - 36 x i + 24 \sqrt{7} x^{2} + 24 \sqrt{7} x (6 i) + 6 \sqrt{7} x + 6 \sqrt{7} (6 i)) i}{4}$

Step 7.14.12.6

Multiply $6$ by $24$ .

$y = \frac{4 x^{4} + 24 x^{3} i + x^{3} + 6 x^{2} i - 4 \sqrt{7} x^{3} - 24 \sqrt{7} x^{2} i - \sqrt{7} x^{2} - 6 \sqrt{7} i x + (- 24 x^{3} - 144 x^{2} i - 6 x^{2} - 36 x i + 24 \sqrt{7} x^{2} + 144 \sqrt{7} x i + 6 \sqrt{7} x + 6 \sqrt{7} (6 i)) i}{4}$

Step 7.14.12.7

Multiply $6$ by $6$ .

Step 7.14.13

Apply the distributive property.

$y = \frac{4 x^{4} + 24 x^{3} i + x^{3} + 6 x^{2} i - 4 \sqrt{7} x^{3} - 24 \sqrt{7} x^{2} i - \sqrt{7} x^{2} - 6 \sqrt{7} i x - 24 x^{3} i - 144 x^{2} i i - 6 x^{2} i - 36 x i i + 24 \sqrt{7} x^{2} i + 144 \sqrt{7} x i i + 6 \sqrt{7} x i + 36 \sqrt{7} i i}{4}$

Step 7.14.14

Simplify.

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Step 7.14.14.1

Multiply $- 144 x^{2} i i$ .

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Step 7.14.14.1.1

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

$y = \frac{4 x^{4} + 24 x^{3} i + x^{3} + 6 x^{2} i - 4 \sqrt{7} x^{3} - 24 \sqrt{7} x^{2} i - \sqrt{7} x^{2} - 6 \sqrt{7} i x - 24 x^{3} i - 144 x^{2} (i i) - 6 x^{2} i - 36 x i i + 24 \sqrt{7} x^{2} i + 144 \sqrt{7} x i i + 6 \sqrt{7} x i + 36 \sqrt{7} i i}{4}$

Step 7.14.14.1.2

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

Step 7.14.14.1.3

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

$y = \frac{4 x^{4} + 24 x^{3} i + x^{3} + 6 x^{2} i - 4 \sqrt{7} x^{3} - 24 \sqrt{7} x^{2} i - \sqrt{7} x^{2} - 6 \sqrt{7} i x - 24 x^{3} i - 144 x^{2} i^{1 + 1} - 6 x^{2} i - 36 x i i + 24 \sqrt{7} x^{2} i + 144 \sqrt{7} x i i + 6 \sqrt{7} x i + 36 \sqrt{7} i i}{4}$

Step 7.14.14.1.4

Add $1$ and $1$ .

$y = \frac{4 x^{4} + 24 x^{3} i + x^{3} + 6 x^{2} i - 4 \sqrt{7} x^{3} - 24 \sqrt{7} x^{2} i - \sqrt{7} x^{2} - 6 \sqrt{7} i x - 24 x^{3} i - 144 x^{2} i^{2} - 6 x^{2} i - 36 x i i + 24 \sqrt{7} x^{2} i + 144 \sqrt{7} x i i + 6 \sqrt{7} x i + 36 \sqrt{7} i i}{4}$

Step 7.14.14.2

Multiply $- 36 x i i$ .

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Step 7.14.14.2.1

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

$y = \frac{4 x^{4} + 24 x^{3} i + x^{3} + 6 x^{2} i - 4 \sqrt{7} x^{3} - 24 \sqrt{7} x^{2} i - \sqrt{7} x^{2} - 6 \sqrt{7} i x - 24 x^{3} i - 144 x^{2} i^{2} - 6 x^{2} i - 36 x (i i) + 24 \sqrt{7} x^{2} i + 144 \sqrt{7} x i i + 6 \sqrt{7} x i + 36 \sqrt{7} i i}{4}$

Step 7.14.14.2.2

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

Step 7.14.14.2.3

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

$y = \frac{4 x^{4} + 24 x^{3} i + x^{3} + 6 x^{2} i - 4 \sqrt{7} x^{3} - 24 \sqrt{7} x^{2} i - \sqrt{7} x^{2} - 6 \sqrt{7} i x - 24 x^{3} i - 144 x^{2} i^{2} - 6 x^{2} i - 36 x i^{1 + 1} + 24 \sqrt{7} x^{2} i + 144 \sqrt{7} x i i + 6 \sqrt{7} x i + 36 \sqrt{7} i i}{4}$

Step 7.14.14.2.4

Add $1$ and $1$ .

$y = \frac{4 x^{4} + 24 x^{3} i + x^{3} + 6 x^{2} i - 4 \sqrt{7} x^{3} - 24 \sqrt{7} x^{2} i - \sqrt{7} x^{2} - 6 \sqrt{7} i x - 24 x^{3} i - 144 x^{2} i^{2} - 6 x^{2} i - 36 x i^{2} + 24 \sqrt{7} x^{2} i + 144 \sqrt{7} x i i + 6 \sqrt{7} x i + 36 \sqrt{7} i i}{4}$

Step 7.14.14.3

Multiply $144 \sqrt{7} x i i$ .

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Step 7.14.14.3.1

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

Step 7.14.14.3.2

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

Step 7.14.14.3.3

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

Step 7.14.14.3.4

Add $1$ and $1$ .

Step 7.14.14.4

Multiply $36 \sqrt{7} i i$ .

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Step 7.14.14.4.1

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

Step 7.14.14.4.2

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

Step 7.14.14.4.3

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

Step 7.14.14.4.4

Add $1$ and $1$ .

Step 7.14.15

Simplify each term.

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Step 7.14.15.1

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

$y = \frac{4 x^{4} + 24 x^{3} i + x^{3} + 6 x^{2} i - 4 \sqrt{7} x^{3} - 24 \sqrt{7} x^{2} i - \sqrt{7} x^{2} - 6 \sqrt{7} i x - 24 x^{3} i - 144 x^{2} \cdot - 1 - 6 x^{2} i - 36 x i^{2} + 24 \sqrt{7} x^{2} i + 144 \sqrt{7} x i^{2} + 6 \sqrt{7} x i + 36 \sqrt{7} i^{2}}{4}$

Step 7.14.15.2

Multiply $- 1$ by $- 144$ .

$y = \frac{4 x^{4} + 24 x^{3} i + x^{3} + 6 x^{2} i - 4 \sqrt{7} x^{3} - 24 \sqrt{7} x^{2} i - \sqrt{7} x^{2} - 6 \sqrt{7} i x - 24 x^{3} i + 144 x^{2} - 6 x^{2} i - 36 x i^{2} + 24 \sqrt{7} x^{2} i + 144 \sqrt{7} x i^{2} + 6 \sqrt{7} x i + 36 \sqrt{7} i^{2}}{4}$

Step 7.14.15.3

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

$y = \frac{4 x^{4} + 24 x^{3} i + x^{3} + 6 x^{2} i - 4 \sqrt{7} x^{3} - 24 \sqrt{7} x^{2} i - \sqrt{7} x^{2} - 6 \sqrt{7} i x - 24 x^{3} i + 144 x^{2} - 6 x^{2} i - 36 x \cdot - 1 + 24 \sqrt{7} x^{2} i + 144 \sqrt{7} x i^{2} + 6 \sqrt{7} x i + 36 \sqrt{7} i^{2}}{4}$

Step 7.14.15.4

Multiply $- 1$ by $- 36$ .

$y = \frac{4 x^{4} + 24 x^{3} i + x^{3} + 6 x^{2} i - 4 \sqrt{7} x^{3} - 24 \sqrt{7} x^{2} i - \sqrt{7} x^{2} - 6 \sqrt{7} i x - 24 x^{3} i + 144 x^{2} - 6 x^{2} i + 36 x + 24 \sqrt{7} x^{2} i + 144 \sqrt{7} x i^{2} + 6 \sqrt{7} x i + 36 \sqrt{7} i^{2}}{4}$

Step 7.14.15.5

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

Step 7.14.15.6

Multiply $- 1$ by $144$ .

$y = \frac{4 x^{4} + 24 x^{3} i + x^{3} + 6 x^{2} i - 4 \sqrt{7} x^{3} - 24 \sqrt{7} x^{2} i - \sqrt{7} x^{2} - 6 \sqrt{7} i x - 24 x^{3} i + 144 x^{2} - 6 x^{2} i + 36 x + 24 \sqrt{7} x^{2} i - 144 \sqrt{7} x + 6 \sqrt{7} x i + 36 \sqrt{7} i^{2}}{4}$

Step 7.14.15.7

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

Step 7.14.15.8

Multiply $- 1$ by $36$ .

Step 7.15

Combine the opposite terms in $4 x^{4} + 24 x^{3} i + x^{3} + 6 x^{2} i - 4 \sqrt{7} x^{3} - 24 \sqrt{7} x^{2} i - \sqrt{7} x^{2} - 6 \sqrt{7} i x - 24 x^{3} i + 144 x^{2} - 6 x^{2} i + 36 x + 24 \sqrt{7} x^{2} i - 144 \sqrt{7} x + 6 \sqrt{7} x i - 36 \sqrt{7}$ .

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Step 7.15.1

Subtract $24 x^{3} i$ from $24 x^{3} i$ .

$y = \frac{4 x^{4} + x^{3} + 6 x^{2} i - 4 \sqrt{7} x^{3} - 24 \sqrt{7} x^{2} i - \sqrt{7} x^{2} - 6 \sqrt{7} i x + 0 + 144 x^{2} - 6 x^{2} i + 36 x + 24 \sqrt{7} x^{2} i - 144 \sqrt{7} x + 6 \sqrt{7} x i - 36 \sqrt{7}}{4}$

Step 7.15.2

Add $4 x^{4} + x^{3} + 6 x^{2} i - 4 \sqrt{7} x^{3} - 24 \sqrt{7} x^{2} i - \sqrt{7} x^{2} - 6 \sqrt{7} i x$ and $0$ .

$y = \frac{4 x^{4} + x^{3} + 6 x^{2} i - 4 \sqrt{7} x^{3} - 24 \sqrt{7} x^{2} i - \sqrt{7} x^{2} - 6 \sqrt{7} i x + 144 x^{2} - 6 x^{2} i + 36 x + 24 \sqrt{7} x^{2} i - 144 \sqrt{7} x + 6 \sqrt{7} x i - 36 \sqrt{7}}{4}$

Step 7.15.3

Subtract $6 x^{2} i$ from $6 x^{2} i$ .

$y = \frac{4 x^{4} + x^{3} - 4 \sqrt{7} x^{3} - 24 \sqrt{7} x^{2} i - \sqrt{7} x^{2} - 6 \sqrt{7} i x + 144 x^{2} + 0 + 36 x + 24 \sqrt{7} x^{2} i - 144 \sqrt{7} x + 6 \sqrt{7} x i - 36 \sqrt{7}}{4}$

Step 7.15.4

Add $4 x^{4} + x^{3} - 4 \sqrt{7} x^{3} - 24 \sqrt{7} x^{2} i - \sqrt{7} x^{2} - 6 \sqrt{7} i x + 144 x^{2}$ and $0$ .

$y = \frac{4 x^{4} + x^{3} - 4 \sqrt{7} x^{3} - 24 \sqrt{7} x^{2} i - \sqrt{7} x^{2} - 6 \sqrt{7} i x + 144 x^{2} + 36 x + 24 \sqrt{7} x^{2} i - 144 \sqrt{7} x + 6 \sqrt{7} x i - 36 \sqrt{7}}{4}$

Step 7.15.5

Add $- 24 \sqrt{7} x^{2} i$ and $24 \sqrt{7} x^{2} i$ .

$y = \frac{4 x^{4} + x^{3} - 4 \sqrt{7} x^{3} - \sqrt{7} x^{2} - 6 \sqrt{7} i x + 144 x^{2} + 36 x + 0 - 144 \sqrt{7} x + 6 \sqrt{7} x i - 36 \sqrt{7}}{4}$

Step 7.15.6

Add $4 x^{4} + x^{3} - 4 \sqrt{7} x^{3} - \sqrt{7} x^{2} - 6 \sqrt{7} i x + 144 x^{2} + 36 x$ and $0$ .

$y = \frac{4 x^{4} + x^{3} - 4 \sqrt{7} x^{3} - \sqrt{7} x^{2} - 6 \sqrt{7} i x + 144 x^{2} + 36 x - 144 \sqrt{7} x + 6 \sqrt{7} x i - 36 \sqrt{7}}{4}$

Step 7.15.7

Reorder the factors in the terms $- 6 \sqrt{7} i x$ and $6 \sqrt{7} x i$ .

$y = \frac{4 x^{4} + x^{3} - 4 \sqrt{7} x^{3} - \sqrt{7} x^{2} - 6 i x \sqrt{7} + 144 x^{2} + 36 x - 144 \sqrt{7} x + 6 i x \sqrt{7} - 36 \sqrt{7}}{4}$

Step 7.15.8

Add $- 6 i x \sqrt{7}$ and $6 i x \sqrt{7}$ .

$y = \frac{4 x^{4} + x^{3} - 4 \sqrt{7} x^{3} - \sqrt{7} x^{2} + 144 x^{2} + 36 x - 144 \sqrt{7} x + 0 - 36 \sqrt{7}}{4}$

Step 7.15.9

Add $4 x^{4} + x^{3} - 4 \sqrt{7} x^{3} - \sqrt{7} x^{2} + 144 x^{2} + 36 x - 144 \sqrt{7} x$ and $0$ .

$y = \frac{4 x^{4} + x^{3} - 4 \sqrt{7} x^{3} - \sqrt{7} x^{2} + 144 x^{2} + 36 x - 144 \sqrt{7} x - 36 \sqrt{7}}{4}$

Step 7.16

Split the fraction $\frac{4 x^{4} + x^{3} - 4 \sqrt{7} x^{3} - \sqrt{7} x^{2} + 144 x^{2} + 36 x - 144 \sqrt{7} x - 36 \sqrt{7}}{4}$ into two fractions.

$y = \frac{4 x^{4} + x^{3} - 4 \sqrt{7} x^{3} - \sqrt{7} x^{2} + 144 x^{2} + 36 x - 144 \sqrt{7} x}{4} + \frac{- 36 \sqrt{7}}{4}$

Step 7.17

Split the fraction $\frac{4 x^{4} + x^{3} - 4 \sqrt{7} x^{3} - \sqrt{7} x^{2} + 144 x^{2} + 36 x - 144 \sqrt{7} x}{4}$ into two fractions.

$y = \frac{4 x^{4} + x^{3} - 4 \sqrt{7} x^{3} - \sqrt{7} x^{2} + 144 x^{2} + 36 x}{4} + \frac{- 144 \sqrt{7} x}{4} + \frac{- 36 \sqrt{7}}{4}$

Step 7.18

Split the fraction $\frac{4 x^{4} + x^{3} - 4 \sqrt{7} x^{3} - \sqrt{7} x^{2} + 144 x^{2} + 36 x}{4}$ into two fractions.

$y = \frac{4 x^{4} + x^{3} - 4 \sqrt{7} x^{3} - \sqrt{7} x^{2} + 144 x^{2}}{4} + \frac{36 x}{4} + \frac{- 144 \sqrt{7} x}{4} + \frac{- 36 \sqrt{7}}{4}$

Step 7.19

Split the fraction $\frac{4 x^{4} + x^{3} - 4 \sqrt{7} x^{3} - \sqrt{7} x^{2} + 144 x^{2}}{4}$ into two fractions.

$y = \frac{4 x^{4} + x^{3} - 4 \sqrt{7} x^{3} - \sqrt{7} x^{2}}{4} + \frac{144 x^{2}}{4} + \frac{36 x}{4} + \frac{- 144 \sqrt{7} x}{4} + \frac{- 36 \sqrt{7}}{4}$

Step 7.20

Split the fraction $\frac{4 x^{4} + x^{3} - 4 \sqrt{7} x^{3} - \sqrt{7} x^{2}}{4}$ into two fractions.

$y = \frac{4 x^{4} + x^{3} - 4 \sqrt{7} x^{3}}{4} + \frac{- \sqrt{7} x^{2}}{4} + \frac{144 x^{2}}{4} + \frac{36 x}{4} + \frac{- 144 \sqrt{7} x}{4} + \frac{- 36 \sqrt{7}}{4}$

Step 7.21

Split the fraction $\frac{4 x^{4} + x^{3} - 4 \sqrt{7} x^{3}}{4}$ into two fractions.

$y = \frac{4 x^{4} + x^{3}}{4} + \frac{- 4 \sqrt{7} x^{3}}{4} + \frac{- \sqrt{7} x^{2}}{4} + \frac{144 x^{2}}{4} + \frac{36 x}{4} + \frac{- 144 \sqrt{7} x}{4} + \frac{- 36 \sqrt{7}}{4}$

Step 7.22

Split the fraction $\frac{4 x^{4} + x^{3}}{4}$ into two fractions.

$y = \frac{4 x^{4}}{4} + \frac{x^{3}}{4} + \frac{- 4 \sqrt{7} x^{3}}{4} + \frac{- \sqrt{7} x^{2}}{4} + \frac{144 x^{2}}{4} + \frac{36 x}{4} + \frac{- 144 \sqrt{7} x}{4} + \frac{- 36 \sqrt{7}}{4}$

Step 7.23

Cancel the common factor of $4$ .

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Step 7.23.1

Cancel the common factor.

$y = \frac{4 x^{4}}{4} + \frac{x^{3}}{4} + \frac{- 4 \sqrt{7} x^{3}}{4} + \frac{- \sqrt{7} x^{2}}{4} + \frac{144 x^{2}}{4} + \frac{36 x}{4} + \frac{- 144 \sqrt{7} x}{4} + \frac{- 36 \sqrt{7}}{4}$

Step 7.23.2

Divide $x^{4}$ by $1$ .

$y = x^{4} + \frac{x^{3}}{4} + \frac{- 4 \sqrt{7} x^{3}}{4} + \frac{- \sqrt{7} x^{2}}{4} + \frac{144 x^{2}}{4} + \frac{36 x}{4} + \frac{- 144 \sqrt{7} x}{4} + \frac{- 36 \sqrt{7}}{4}$

Step 7.24

Cancel the common factor of $- 4$ and $4$ .

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Step 7.24.1

Factor $4$ out of $- 4 \sqrt{7} x^{3}$ .

$y = x^{4} + \frac{x^{3}}{4} + \frac{4 (- \sqrt{7} x^{3})}{4} + \frac{- \sqrt{7} x^{2}}{4} + \frac{144 x^{2}}{4} + \frac{36 x}{4} + \frac{- 144 \sqrt{7} x}{4} + \frac{- 36 \sqrt{7}}{4}$

Step 7.24.2

Cancel the common factors.

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Step 7.24.2.1

Factor $4$ out of $4$ .

$y = x^{4} + \frac{x^{3}}{4} + \frac{4 (- \sqrt{7} x^{3})}{4 (1)} + \frac{- \sqrt{7} x^{2}}{4} + \frac{144 x^{2}}{4} + \frac{36 x}{4} + \frac{- 144 \sqrt{7} x}{4} + \frac{- 36 \sqrt{7}}{4}$

Step 7.24.2.2

Cancel the common factor.

$y = x^{4} + \frac{x^{3}}{4} + \frac{4 (- \sqrt{7} x^{3})}{4 \cdot 1} + \frac{- \sqrt{7} x^{2}}{4} + \frac{144 x^{2}}{4} + \frac{36 x}{4} + \frac{- 144 \sqrt{7} x}{4} + \frac{- 36 \sqrt{7}}{4}$

Step 7.24.2.3

Rewrite the expression.

$y = x^{4} + \frac{x^{3}}{4} + \frac{- \sqrt{7} x^{3}}{1} + \frac{- \sqrt{7} x^{2}}{4} + \frac{144 x^{2}}{4} + \frac{36 x}{4} + \frac{- 144 \sqrt{7} x}{4} + \frac{- 36 \sqrt{7}}{4}$

Step 7.24.2.4

Divide $- \sqrt{7} x^{3}$ by $1$ .

$y = x^{4} + \frac{x^{3}}{4} - \sqrt{7} x^{3} + \frac{- \sqrt{7} x^{2}}{4} + \frac{144 x^{2}}{4} + \frac{36 x}{4} + \frac{- 144 \sqrt{7} x}{4} + \frac{- 36 \sqrt{7}}{4}$

Step 7.25

Move the negative in front of the fraction.

$y = x^{4} + \frac{x^{3}}{4} - \sqrt{7} x^{3} - \frac{\sqrt{7} x^{2}}{4} + \frac{144 x^{2}}{4} + \frac{36 x}{4} + \frac{- 144 \sqrt{7} x}{4} + \frac{- 36 \sqrt{7}}{4}$

Step 7.26

Cancel the common factor of $144$ and $4$ .

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Step 7.26.1

Factor $4$ out of $144 x^{2}$ .

$y = x^{4} + \frac{x^{3}}{4} - \sqrt{7} x^{3} - \frac{\sqrt{7} x^{2}}{4} + \frac{4 (36 x^{2})}{4} + \frac{36 x}{4} + \frac{- 144 \sqrt{7} x}{4} + \frac{- 36 \sqrt{7}}{4}$

Step 7.26.2

Cancel the common factors.

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Step 7.26.2.1

Factor $4$ out of $4$ .

$y = x^{4} + \frac{x^{3}}{4} - \sqrt{7} x^{3} - \frac{\sqrt{7} x^{2}}{4} + \frac{4 (36 x^{2})}{4 (1)} + \frac{36 x}{4} + \frac{- 144 \sqrt{7} x}{4} + \frac{- 36 \sqrt{7}}{4}$

Step 7.26.2.2

Cancel the common factor.

$y = x^{4} + \frac{x^{3}}{4} - \sqrt{7} x^{3} - \frac{\sqrt{7} x^{2}}{4} + \frac{4 (36 x^{2})}{4 \cdot 1} + \frac{36 x}{4} + \frac{- 144 \sqrt{7} x}{4} + \frac{- 36 \sqrt{7}}{4}$

Step 7.26.2.3

Rewrite the expression.

$y = x^{4} + \frac{x^{3}}{4} - \sqrt{7} x^{3} - \frac{\sqrt{7} x^{2}}{4} + \frac{36 x^{2}}{1} + \frac{36 x}{4} + \frac{- 144 \sqrt{7} x}{4} + \frac{- 36 \sqrt{7}}{4}$

Step 7.26.2.4

Divide $36 x^{2}$ by $1$ .

$y = x^{4} + \frac{x^{3}}{4} - \sqrt{7} x^{3} - \frac{\sqrt{7} x^{2}}{4} + 36 x^{2} + \frac{36 x}{4} + \frac{- 144 \sqrt{7} x}{4} + \frac{- 36 \sqrt{7}}{4}$

Step 7.27

Cancel the common factor of $36$ and $4$ .

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Step 7.27.1

Factor $4$ out of $36 x$ .

$y = x^{4} + \frac{x^{3}}{4} - \sqrt{7} x^{3} - \frac{\sqrt{7} x^{2}}{4} + 36 x^{2} + \frac{4 (9 x)}{4} + \frac{- 144 \sqrt{7} x}{4} + \frac{- 36 \sqrt{7}}{4}$

Step 7.27.2

Cancel the common factors.

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Step 7.27.2.1

Factor $4$ out of $4$ .

$y = x^{4} + \frac{x^{3}}{4} - \sqrt{7} x^{3} - \frac{\sqrt{7} x^{2}}{4} + 36 x^{2} + \frac{4 (9 x)}{4 (1)} + \frac{- 144 \sqrt{7} x}{4} + \frac{- 36 \sqrt{7}}{4}$

Step 7.27.2.2

Cancel the common factor.

$y = x^{4} + \frac{x^{3}}{4} - \sqrt{7} x^{3} - \frac{\sqrt{7} x^{2}}{4} + 36 x^{2} + \frac{4 (9 x)}{4 \cdot 1} + \frac{- 144 \sqrt{7} x}{4} + \frac{- 36 \sqrt{7}}{4}$

Step 7.27.2.3

Rewrite the expression.

$y = x^{4} + \frac{x^{3}}{4} - \sqrt{7} x^{3} - \frac{\sqrt{7} x^{2}}{4} + 36 x^{2} + \frac{9 x}{1} + \frac{- 144 \sqrt{7} x}{4} + \frac{- 36 \sqrt{7}}{4}$

Step 7.27.2.4

Divide $9 x$ by $1$ .

$y = x^{4} + \frac{x^{3}}{4} - \sqrt{7} x^{3} - \frac{\sqrt{7} x^{2}}{4} + 36 x^{2} + 9 x + \frac{- 144 \sqrt{7} x}{4} + \frac{- 36 \sqrt{7}}{4}$

Step 7.28

Cancel the common factor of $- 144$ and $4$ .

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Step 7.28.1

Factor $4$ out of $- 144 \sqrt{7} x$ .

$y = x^{4} + \frac{x^{3}}{4} - \sqrt{7} x^{3} - \frac{\sqrt{7} x^{2}}{4} + 36 x^{2} + 9 x + \frac{4 (- 36 \sqrt{7} x)}{4} + \frac{- 36 \sqrt{7}}{4}$

Step 7.28.2

Cancel the common factors.

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Step 7.28.2.1

Factor $4$ out of $4$ .

$y = x^{4} + \frac{x^{3}}{4} - \sqrt{7} x^{3} - \frac{\sqrt{7} x^{2}}{4} + 36 x^{2} + 9 x + \frac{4 (- 36 \sqrt{7} x)}{4 (1)} + \frac{- 36 \sqrt{7}}{4}$

Step 7.28.2.2

Cancel the common factor.

$y = x^{4} + \frac{x^{3}}{4} - \sqrt{7} x^{3} - \frac{\sqrt{7} x^{2}}{4} + 36 x^{2} + 9 x + \frac{4 (- 36 \sqrt{7} x)}{4 \cdot 1} + \frac{- 36 \sqrt{7}}{4}$

Step 7.28.2.3

Rewrite the expression.

$y = x^{4} + \frac{x^{3}}{4} - \sqrt{7} x^{3} - \frac{\sqrt{7} x^{2}}{4} + 36 x^{2} + 9 x + \frac{- 36 \sqrt{7} x}{1} + \frac{- 36 \sqrt{7}}{4}$

Step 7.28.2.4

Divide $- 36 \sqrt{7} x$ by $1$ .

$y = x^{4} + \frac{x^{3}}{4} - \sqrt{7} x^{3} - \frac{\sqrt{7} x^{2}}{4} + 36 x^{2} + 9 x - 36 \sqrt{7} x + \frac{- 36 \sqrt{7}}{4}$

Step 7.29

Cancel the common factor of $- 36$ and $4$ .

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Step 7.29.1

Factor $4$ out of $- 36 \sqrt{7}$ .

$y = x^{4} + \frac{x^{3}}{4} - \sqrt{7} x^{3} - \frac{\sqrt{7} x^{2}}{4} + 36 x^{2} + 9 x - 36 \sqrt{7} x + \frac{4 (- 9 \sqrt{7})}{4}$

Step 7.29.2

Cancel the common factors.

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Step 7.29.2.1

Factor $4$ out of $4$ .

$y = x^{4} + \frac{x^{3}}{4} - \sqrt{7} x^{3} - \frac{\sqrt{7} x^{2}}{4} + 36 x^{2} + 9 x - 36 \sqrt{7} x + \frac{4 (- 9 \sqrt{7})}{4 (1)}$

Step 7.29.2.2

Cancel the common factor.

$y = x^{4} + \frac{x^{3}}{4} - \sqrt{7} x^{3} - \frac{\sqrt{7} x^{2}}{4} + 36 x^{2} + 9 x - 36 \sqrt{7} x + \frac{4 (- 9 \sqrt{7})}{4 \cdot 1}$

Step 7.29.2.3

Rewrite the expression.

$y = x^{4} + \frac{x^{3}}{4} - \sqrt{7} x^{3} - \frac{\sqrt{7} x^{2}}{4} + 36 x^{2} + 9 x - 36 \sqrt{7} x + \frac{- 9 \sqrt{7}}{1}$

Step 7.29.2.4

Divide $- 9 \sqrt{7}$ by $1$ .

$y = x^{4} + \frac{x^{3}}{4} - \sqrt{7} x^{3} - \frac{\sqrt{7} x^{2}}{4} + 36 x^{2} + 9 x - 36 \sqrt{7} x - 9 \sqrt{7}$

Step 8