Precalculus Examples

$- 3$ , $- 3$ , $2 - 5 i$

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 $x = - 3$ was found by solving for $x$ when $x - (- 3) = y$ and $y = 0$ .

The factor is $x + 3$

Step 3

The root at $x = 2 - 5 i$ was found by solving for $x$ when $x - (2 - 5 i) = y$ and $y = 0$ .

The factor is $x - 2 + 5 i$

Step 4

The root at $x = 2 + 5 i$ was found by solving for $x$ when $x - (2 + 5 i) = y$ and $y = 0$ .

The factor is $x - 2 - 5 i$

Step 5

Combine all the factors into a single equation.

$y = (x + 3) (x + 3) (x - 2 + 5 i) (x - 2 - 5 i)$

Step 6

Multiply all the factors to simplify the equation $y = (x + 3) (x + 3) (x - 2 + 5 i) (x - 2 - 5 i)$ .

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Step 6.1

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

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Step 6.1.1

Apply the distributive property.

$y = (x (x + 3) + 3 (x + 3)) (x - 2 + 5 i) (x - 2 - 5 i)$

Step 6.1.2

Apply the distributive property.

$y = (x \cdot x + x \cdot 3 + 3 (x + 3)) (x - 2 + 5 i) (x - 2 - 5 i)$

Step 6.1.3

Apply the distributive property.

$y = (x \cdot x + x \cdot 3 + 3 x + 3 \cdot 3) (x - 2 + 5 i) (x - 2 - 5 i)$

Step 6.2

Simplify and combine like terms.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Multiply $x$ by $x$ .

$y = (x^{2} + x \cdot 3 + 3 x + 3 \cdot 3) (x - 2 + 5 i) (x - 2 - 5 i)$

Step 6.2.1.2

Move $3$ to the left of $x$ .

$y = (x^{2} + 3 \cdot x + 3 x + 3 \cdot 3) (x - 2 + 5 i) (x - 2 - 5 i)$

Step 6.2.1.3

Multiply $3$ by $3$ .

$y = (x^{2} + 3 x + 3 x + 9) (x - 2 + 5 i) (x - 2 - 5 i)$

Step 6.2.2

Add $3 x$ and $3 x$ .

$y = (x^{2} + 6 x + 9) (x - 2 + 5 i) (x - 2 - 5 i)$

Step 6.3

Expand $(x^{2} + 6 x + 9) (x - 2 + 5 i)$ by multiplying each term in the first expression by each term in the second expression.

$y = (x^{2} x + x^{2} \cdot - 2 + x^{2} (5 i) + 6 x \cdot x + 6 x \cdot - 2 + 6 x (5 i) + 9 x + 9 \cdot - 2 + 9 (5 i)) (x - 2 - 5 i)$

Step 6.4

Simplify terms.

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Step 6.4.1

Simplify each term.

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Step 6.4.1.1

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

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Step 6.4.1.1.1

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

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Step 6.4.1.1.1.1

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

$y = (x^{2} x + x^{2} \cdot - 2 + x^{2} (5 i) + 6 x \cdot x + 6 x \cdot - 2 + 6 x (5 i) + 9 x + 9 \cdot - 2 + 9 (5 i)) (x - 2 - 5 i)$

Step 6.4.1.1.1.2

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

$y = (x^{2 + 1} + x^{2} \cdot - 2 + x^{2} (5 i) + 6 x \cdot x + 6 x \cdot - 2 + 6 x (5 i) + 9 x + 9 \cdot - 2 + 9 (5 i)) (x - 2 - 5 i)$

Step 6.4.1.1.2

Add $2$ and $1$ .

$y = (x^{3} + x^{2} \cdot - 2 + x^{2} (5 i) + 6 x \cdot x + 6 x \cdot - 2 + 6 x (5 i) + 9 x + 9 \cdot - 2 + 9 (5 i)) (x - 2 - 5 i)$

Step 6.4.1.2

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

$y = (x^{3} - 2 \cdot x^{2} + x^{2} (5 i) + 6 x \cdot x + 6 x \cdot - 2 + 6 x (5 i) + 9 x + 9 \cdot - 2 + 9 (5 i)) (x - 2 - 5 i)$

Step 6.4.1.3

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

$y = (x^{3} - 2 x^{2} + 5 \cdot (x^{2} i) + 6 x \cdot x + 6 x \cdot - 2 + 6 x (5 i) + 9 x + 9 \cdot - 2 + 9 (5 i)) (x - 2 - 5 i)$

Step 6.4.1.4

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

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Step 6.4.1.4.1

Move $x$ .

$y = (x^{3} - 2 x^{2} + 5 x^{2} i + 6 (x \cdot x) + 6 x \cdot - 2 + 6 x (5 i) + 9 x + 9 \cdot - 2 + 9 (5 i)) (x - 2 - 5 i)$

Step 6.4.1.4.2

Multiply $x$ by $x$ .

$y = (x^{3} - 2 x^{2} + 5 x^{2} i + 6 x^{2} + 6 x \cdot - 2 + 6 x (5 i) + 9 x + 9 \cdot - 2 + 9 (5 i)) (x - 2 - 5 i)$

Step 6.4.1.5

Multiply $- 2$ by $6$ .

$y = (x^{3} - 2 x^{2} + 5 x^{2} i + 6 x^{2} - 12 x + 6 x (5 i) + 9 x + 9 \cdot - 2 + 9 (5 i)) (x - 2 - 5 i)$

Step 6.4.1.6

Multiply $5$ by $6$ .

$y = (x^{3} - 2 x^{2} + 5 x^{2} i + 6 x^{2} - 12 x + 30 x i + 9 x + 9 \cdot - 2 + 9 (5 i)) (x - 2 - 5 i)$

Step 6.4.1.7

Multiply $9$ by $- 2$ .

$y = (x^{3} - 2 x^{2} + 5 x^{2} i + 6 x^{2} - 12 x + 30 x i + 9 x - 18 + 9 (5 i)) (x - 2 - 5 i)$

Step 6.4.1.8

Multiply $5$ by $9$ .

$y = (x^{3} - 2 x^{2} + 5 x^{2} i + 6 x^{2} - 12 x + 30 x i + 9 x - 18 + 45 i) (x - 2 - 5 i)$

Step 6.4.2

Simplify by adding terms.

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Step 6.4.2.1

Add $- 2 x^{2}$ and $6 x^{2}$ .

$y = (x^{3} + 4 x^{2} + 5 x^{2} i - 12 x + 30 x i + 9 x - 18 + 45 i) (x - 2 - 5 i)$

Step 6.4.2.2

Add $- 12 x$ and $9 x$ .

$y = (x^{3} + 4 x^{2} + 5 x^{2} i + 30 x i - 3 x - 18 + 45 i) (x - 2 - 5 i)$

Step 6.5

Expand $(x^{3} + 4 x^{2} + 5 x^{2} i + 30 x i - 3 x - 18 + 45 i) (x - 2 - 5 i)$ by multiplying each term in the first expression by each term in the second expression.

$y = x^{3} x + x^{3} \cdot - 2 + x^{3} (- 5 i) + 4 x^{2} x + 4 x^{2} \cdot - 2 + 4 x^{2} (- 5 i) + 5 x^{2} i x + 5 x^{2} i \cdot - 2 + 5 x^{2} i (- 5 i) + 30 x i x + 30 x i \cdot - 2 + 30 x i (- 5 i) - 3 x \cdot x - 3 x \cdot - 2 - 3 x (- 5 i) - 18 x - 18 \cdot - 2 - 18 (- 5 i) + 45 i x + 45 i \cdot - 2 + 45 i (- 5 i)$

Step 6.6

Simplify terms.

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Step 6.6.1

Simplify each term.

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Step 6.6.1.1

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

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Step 6.6.1.1.1

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

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Step 6.6.1.1.1.1

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

Step 6.6.1.1.1.2

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

$y = x^{3 + 1} + x^{3} \cdot - 2 + x^{3} (- 5 i) + 4 x^{2} x + 4 x^{2} \cdot - 2 + 4 x^{2} (- 5 i) + 5 x^{2} i x + 5 x^{2} i \cdot - 2 + 5 x^{2} i (- 5 i) + 30 x i x + 30 x i \cdot - 2 + 30 x i (- 5 i) - 3 x \cdot x - 3 x \cdot - 2 - 3 x (- 5 i) - 18 x - 18 \cdot - 2 - 18 (- 5 i) + 45 i x + 45 i \cdot - 2 + 45 i (- 5 i)$

Step 6.6.1.1.2

Add $3$ and $1$ .

$y = x^{4} + x^{3} \cdot - 2 + x^{3} (- 5 i) + 4 x^{2} x + 4 x^{2} \cdot - 2 + 4 x^{2} (- 5 i) + 5 x^{2} i x + 5 x^{2} i \cdot - 2 + 5 x^{2} i (- 5 i) + 30 x i x + 30 x i \cdot - 2 + 30 x i (- 5 i) - 3 x \cdot x - 3 x \cdot - 2 - 3 x (- 5 i) - 18 x - 18 \cdot - 2 - 18 (- 5 i) + 45 i x + 45 i \cdot - 2 + 45 i (- 5 i)$

Step 6.6.1.2

Move $- 2$ to the left of $x^{3}$ .

$y = x^{4} - 2 \cdot x^{3} + x^{3} (- 5 i) + 4 x^{2} x + 4 x^{2} \cdot - 2 + 4 x^{2} (- 5 i) + 5 x^{2} i x + 5 x^{2} i \cdot - 2 + 5 x^{2} i (- 5 i) + 30 x i x + 30 x i \cdot - 2 + 30 x i (- 5 i) - 3 x \cdot x - 3 x \cdot - 2 - 3 x (- 5 i) - 18 x - 18 \cdot - 2 - 18 (- 5 i) + 45 i x + 45 i \cdot - 2 + 45 i (- 5 i)$

Step 6.6.1.3

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

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Step 6.6.1.3.1

Move $x$ .

$y = x^{4} - 2 x^{3} + x^{3} (- 5 i) + 4 (x \cdot x^{2}) + 4 x^{2} \cdot - 2 + 4 x^{2} (- 5 i) + 5 x^{2} i x + 5 x^{2} i \cdot - 2 + 5 x^{2} i (- 5 i) + 30 x i x + 30 x i \cdot - 2 + 30 x i (- 5 i) - 3 x \cdot x - 3 x \cdot - 2 - 3 x (- 5 i) - 18 x - 18 \cdot - 2 - 18 (- 5 i) + 45 i x + 45 i \cdot - 2 + 45 i (- 5 i)$

Step 6.6.1.3.2

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

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Step 6.6.1.3.2.1

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

Step 6.6.1.3.2.2

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

$y = x^{4} - 2 x^{3} + x^{3} (- 5 i) + 4 x^{1 + 2} + 4 x^{2} \cdot - 2 + 4 x^{2} (- 5 i) + 5 x^{2} i x + 5 x^{2} i \cdot - 2 + 5 x^{2} i (- 5 i) + 30 x i x + 30 x i \cdot - 2 + 30 x i (- 5 i) - 3 x \cdot x - 3 x \cdot - 2 - 3 x (- 5 i) - 18 x - 18 \cdot - 2 - 18 (- 5 i) + 45 i x + 45 i \cdot - 2 + 45 i (- 5 i)$

Step 6.6.1.3.3

Add $1$ and $2$ .

$y = x^{4} - 2 x^{3} + x^{3} (- 5 i) + 4 x^{3} + 4 x^{2} \cdot - 2 + 4 x^{2} (- 5 i) + 5 x^{2} i x + 5 x^{2} i \cdot - 2 + 5 x^{2} i (- 5 i) + 30 x i x + 30 x i \cdot - 2 + 30 x i (- 5 i) - 3 x \cdot x - 3 x \cdot - 2 - 3 x (- 5 i) - 18 x - 18 \cdot - 2 - 18 (- 5 i) + 45 i x + 45 i \cdot - 2 + 45 i (- 5 i)$

Step 6.6.1.4

Multiply $- 2$ by $4$ .

$y = x^{4} - 2 x^{3} + x^{3} (- 5 i) + 4 x^{3} - 8 x^{2} + 4 x^{2} (- 5 i) + 5 x^{2} i x + 5 x^{2} i \cdot - 2 + 5 x^{2} i (- 5 i) + 30 x i x + 30 x i \cdot - 2 + 30 x i (- 5 i) - 3 x \cdot x - 3 x \cdot - 2 - 3 x (- 5 i) - 18 x - 18 \cdot - 2 - 18 (- 5 i) + 45 i x + 45 i \cdot - 2 + 45 i (- 5 i)$

Step 6.6.1.5

Multiply $- 5$ by $4$ .

$y = x^{4} - 2 x^{3} + x^{3} (- 5 i) + 4 x^{3} - 8 x^{2} - 20 x^{2} i + 5 x^{2} i x + 5 x^{2} i \cdot - 2 + 5 x^{2} i (- 5 i) + 30 x i x + 30 x i \cdot - 2 + 30 x i (- 5 i) - 3 x \cdot x - 3 x \cdot - 2 - 3 x (- 5 i) - 18 x - 18 \cdot - 2 - 18 (- 5 i) + 45 i x + 45 i \cdot - 2 + 45 i (- 5 i)$

Step 6.6.1.6

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

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Step 6.6.1.6.1

Move $x$ .

$y = x^{4} - 2 x^{3} + x^{3} (- 5 i) + 4 x^{3} - 8 x^{2} - 20 x^{2} i + 5 (x \cdot x^{2}) i + 5 x^{2} i \cdot - 2 + 5 x^{2} i (- 5 i) + 30 x i x + 30 x i \cdot - 2 + 30 x i (- 5 i) - 3 x \cdot x - 3 x \cdot - 2 - 3 x (- 5 i) - 18 x - 18 \cdot - 2 - 18 (- 5 i) + 45 i x + 45 i \cdot - 2 + 45 i (- 5 i)$

Step 6.6.1.6.2

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

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Step 6.6.1.6.2.1

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

Step 6.6.1.6.2.2

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

$y = x^{4} - 2 x^{3} + x^{3} (- 5 i) + 4 x^{3} - 8 x^{2} - 20 x^{2} i + 5 x^{1 + 2} i + 5 x^{2} i \cdot - 2 + 5 x^{2} i (- 5 i) + 30 x i x + 30 x i \cdot - 2 + 30 x i (- 5 i) - 3 x \cdot x - 3 x \cdot - 2 - 3 x (- 5 i) - 18 x - 18 \cdot - 2 - 18 (- 5 i) + 45 i x + 45 i \cdot - 2 + 45 i (- 5 i)$

Step 6.6.1.6.3

Add $1$ and $2$ .

$y = x^{4} - 2 x^{3} + x^{3} (- 5 i) + 4 x^{3} - 8 x^{2} - 20 x^{2} i + 5 x^{3} i + 5 x^{2} i \cdot - 2 + 5 x^{2} i (- 5 i) + 30 x i x + 30 x i \cdot - 2 + 30 x i (- 5 i) - 3 x \cdot x - 3 x \cdot - 2 - 3 x (- 5 i) - 18 x - 18 \cdot - 2 - 18 (- 5 i) + 45 i x + 45 i \cdot - 2 + 45 i (- 5 i)$

Step 6.6.1.7

Multiply $- 2$ by $5$ .

$y = x^{4} - 2 x^{3} + x^{3} (- 5 i) + 4 x^{3} - 8 x^{2} - 20 x^{2} i + 5 x^{3} i - 10 x^{2} i + 5 x^{2} i (- 5 i) + 30 x i x + 30 x i \cdot - 2 + 30 x i (- 5 i) - 3 x \cdot x - 3 x \cdot - 2 - 3 x (- 5 i) - 18 x - 18 \cdot - 2 - 18 (- 5 i) + 45 i x + 45 i \cdot - 2 + 45 i (- 5 i)$

Step 6.6.1.8

Multiply $5 x^{2} i (- 5 i)$ .

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Step 6.6.1.8.1

Multiply $- 5$ by $5$ .

$y = x^{4} - 2 x^{3} + x^{3} (- 5 i) + 4 x^{3} - 8 x^{2} - 20 x^{2} i + 5 x^{3} i - 10 x^{2} i - 25 x^{2} i i + 30 x i x + 30 x i \cdot - 2 + 30 x i (- 5 i) - 3 x \cdot x - 3 x \cdot - 2 - 3 x (- 5 i) - 18 x - 18 \cdot - 2 - 18 (- 5 i) + 45 i x + 45 i \cdot - 2 + 45 i (- 5 i)$

Step 6.6.1.8.2

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

$y = x^{4} - 2 x^{3} + x^{3} (- 5 i) + 4 x^{3} - 8 x^{2} - 20 x^{2} i + 5 x^{3} i - 10 x^{2} i - 25 x^{2} (i i) + 30 x i x + 30 x i \cdot - 2 + 30 x i (- 5 i) - 3 x \cdot x - 3 x \cdot - 2 - 3 x (- 5 i) - 18 x - 18 \cdot - 2 - 18 (- 5 i) + 45 i x + 45 i \cdot - 2 + 45 i (- 5 i)$

Step 6.6.1.8.3

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

Step 6.6.1.8.4

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

$y = x^{4} - 2 x^{3} + x^{3} (- 5 i) + 4 x^{3} - 8 x^{2} - 20 x^{2} i + 5 x^{3} i - 10 x^{2} i - 25 x^{2} i^{1 + 1} + 30 x i x + 30 x i \cdot - 2 + 30 x i (- 5 i) - 3 x \cdot x - 3 x \cdot - 2 - 3 x (- 5 i) - 18 x - 18 \cdot - 2 - 18 (- 5 i) + 45 i x + 45 i \cdot - 2 + 45 i (- 5 i)$

Step 6.6.1.8.5

Add $1$ and $1$ .

$y = x^{4} - 2 x^{3} + x^{3} (- 5 i) + 4 x^{3} - 8 x^{2} - 20 x^{2} i + 5 x^{3} i - 10 x^{2} i - 25 x^{2} i^{2} + 30 x i x + 30 x i \cdot - 2 + 30 x i (- 5 i) - 3 x \cdot x - 3 x \cdot - 2 - 3 x (- 5 i) - 18 x - 18 \cdot - 2 - 18 (- 5 i) + 45 i x + 45 i \cdot - 2 + 45 i (- 5 i)$

Step 6.6.1.9

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

$y = x^{4} - 2 x^{3} + x^{3} (- 5 i) + 4 x^{3} - 8 x^{2} - 20 x^{2} i + 5 x^{3} i - 10 x^{2} i - 25 x^{2} \cdot - 1 + 30 x i x + 30 x i \cdot - 2 + 30 x i (- 5 i) - 3 x \cdot x - 3 x \cdot - 2 - 3 x (- 5 i) - 18 x - 18 \cdot - 2 - 18 (- 5 i) + 45 i x + 45 i \cdot - 2 + 45 i (- 5 i)$

Step 6.6.1.10

Multiply $- 1$ by $- 25$ .

$y = x^{4} - 2 x^{3} + x^{3} (- 5 i) + 4 x^{3} - 8 x^{2} - 20 x^{2} i + 5 x^{3} i - 10 x^{2} i + 25 x^{2} + 30 x i x + 30 x i \cdot - 2 + 30 x i (- 5 i) - 3 x \cdot x - 3 x \cdot - 2 - 3 x (- 5 i) - 18 x - 18 \cdot - 2 - 18 (- 5 i) + 45 i x + 45 i \cdot - 2 + 45 i (- 5 i)$

Step 6.6.1.11

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

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Step 6.6.1.11.1

Move $x$ .

$y = x^{4} - 2 x^{3} + x^{3} (- 5 i) + 4 x^{3} - 8 x^{2} - 20 x^{2} i + 5 x^{3} i - 10 x^{2} i + 25 x^{2} + 30 (x \cdot x) i + 30 x i \cdot - 2 + 30 x i (- 5 i) - 3 x \cdot x - 3 x \cdot - 2 - 3 x (- 5 i) - 18 x - 18 \cdot - 2 - 18 (- 5 i) + 45 i x + 45 i \cdot - 2 + 45 i (- 5 i)$

Step 6.6.1.11.2

Multiply $x$ by $x$ .

$y = x^{4} - 2 x^{3} + x^{3} (- 5 i) + 4 x^{3} - 8 x^{2} - 20 x^{2} i + 5 x^{3} i - 10 x^{2} i + 25 x^{2} + 30 x^{2} i + 30 x i \cdot - 2 + 30 x i (- 5 i) - 3 x \cdot x - 3 x \cdot - 2 - 3 x (- 5 i) - 18 x - 18 \cdot - 2 - 18 (- 5 i) + 45 i x + 45 i \cdot - 2 + 45 i (- 5 i)$

Step 6.6.1.12

Multiply $- 2$ by $30$ .

$y = x^{4} - 2 x^{3} + x^{3} (- 5 i) + 4 x^{3} - 8 x^{2} - 20 x^{2} i + 5 x^{3} i - 10 x^{2} i + 25 x^{2} + 30 x^{2} i - 60 x i + 30 x i (- 5 i) - 3 x \cdot x - 3 x \cdot - 2 - 3 x (- 5 i) - 18 x - 18 \cdot - 2 - 18 (- 5 i) + 45 i x + 45 i \cdot - 2 + 45 i (- 5 i)$

Step 6.6.1.13

Multiply $30 x i (- 5 i)$ .

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Step 6.6.1.13.1

Multiply $- 5$ by $30$ .

$y = x^{4} - 2 x^{3} + x^{3} (- 5 i) + 4 x^{3} - 8 x^{2} - 20 x^{2} i + 5 x^{3} i - 10 x^{2} i + 25 x^{2} + 30 x^{2} i - 60 x i - 150 x i i - 3 x \cdot x - 3 x \cdot - 2 - 3 x (- 5 i) - 18 x - 18 \cdot - 2 - 18 (- 5 i) + 45 i x + 45 i \cdot - 2 + 45 i (- 5 i)$

Step 6.6.1.13.2

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

$y = x^{4} - 2 x^{3} + x^{3} (- 5 i) + 4 x^{3} - 8 x^{2} - 20 x^{2} i + 5 x^{3} i - 10 x^{2} i + 25 x^{2} + 30 x^{2} i - 60 x i - 150 x (i i) - 3 x \cdot x - 3 x \cdot - 2 - 3 x (- 5 i) - 18 x - 18 \cdot - 2 - 18 (- 5 i) + 45 i x + 45 i \cdot - 2 + 45 i (- 5 i)$

Step 6.6.1.13.3

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

Step 6.6.1.13.4

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

$y = x^{4} - 2 x^{3} + x^{3} (- 5 i) + 4 x^{3} - 8 x^{2} - 20 x^{2} i + 5 x^{3} i - 10 x^{2} i + 25 x^{2} + 30 x^{2} i - 60 x i - 150 x i^{1 + 1} - 3 x \cdot x - 3 x \cdot - 2 - 3 x (- 5 i) - 18 x - 18 \cdot - 2 - 18 (- 5 i) + 45 i x + 45 i \cdot - 2 + 45 i (- 5 i)$

Step 6.6.1.13.5

Add $1$ and $1$ .

$y = x^{4} - 2 x^{3} + x^{3} (- 5 i) + 4 x^{3} - 8 x^{2} - 20 x^{2} i + 5 x^{3} i - 10 x^{2} i + 25 x^{2} + 30 x^{2} i - 60 x i - 150 x i^{2} - 3 x \cdot x - 3 x \cdot - 2 - 3 x (- 5 i) - 18 x - 18 \cdot - 2 - 18 (- 5 i) + 45 i x + 45 i \cdot - 2 + 45 i (- 5 i)$

Step 6.6.1.14

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

$y = x^{4} - 2 x^{3} + x^{3} (- 5 i) + 4 x^{3} - 8 x^{2} - 20 x^{2} i + 5 x^{3} i - 10 x^{2} i + 25 x^{2} + 30 x^{2} i - 60 x i - 150 x \cdot - 1 - 3 x \cdot x - 3 x \cdot - 2 - 3 x (- 5 i) - 18 x - 18 \cdot - 2 - 18 (- 5 i) + 45 i x + 45 i \cdot - 2 + 45 i (- 5 i)$

Step 6.6.1.15

Multiply $- 1$ by $- 150$ .

$y = x^{4} - 2 x^{3} + x^{3} (- 5 i) + 4 x^{3} - 8 x^{2} - 20 x^{2} i + 5 x^{3} i - 10 x^{2} i + 25 x^{2} + 30 x^{2} i - 60 x i + 150 x - 3 x \cdot x - 3 x \cdot - 2 - 3 x (- 5 i) - 18 x - 18 \cdot - 2 - 18 (- 5 i) + 45 i x + 45 i \cdot - 2 + 45 i (- 5 i)$

Step 6.6.1.16

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

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Step 6.6.1.16.1

Move $x$ .

$y = x^{4} - 2 x^{3} + x^{3} (- 5 i) + 4 x^{3} - 8 x^{2} - 20 x^{2} i + 5 x^{3} i - 10 x^{2} i + 25 x^{2} + 30 x^{2} i - 60 x i + 150 x - 3 (x \cdot x) - 3 x \cdot - 2 - 3 x (- 5 i) - 18 x - 18 \cdot - 2 - 18 (- 5 i) + 45 i x + 45 i \cdot - 2 + 45 i (- 5 i)$

Step 6.6.1.16.2

Multiply $x$ by $x$ .

$y = x^{4} - 2 x^{3} + x^{3} (- 5 i) + 4 x^{3} - 8 x^{2} - 20 x^{2} i + 5 x^{3} i - 10 x^{2} i + 25 x^{2} + 30 x^{2} i - 60 x i + 150 x - 3 x^{2} - 3 x \cdot - 2 - 3 x (- 5 i) - 18 x - 18 \cdot - 2 - 18 (- 5 i) + 45 i x + 45 i \cdot - 2 + 45 i (- 5 i)$

Step 6.6.1.17

Multiply $- 2$ by $- 3$ .

$y = x^{4} - 2 x^{3} + x^{3} (- 5 i) + 4 x^{3} - 8 x^{2} - 20 x^{2} i + 5 x^{3} i - 10 x^{2} i + 25 x^{2} + 30 x^{2} i - 60 x i + 150 x - 3 x^{2} + 6 x - 3 x (- 5 i) - 18 x - 18 \cdot - 2 - 18 (- 5 i) + 45 i x + 45 i \cdot - 2 + 45 i (- 5 i)$

Step 6.6.1.18

Multiply $- 5$ by $- 3$ .

$y = x^{4} - 2 x^{3} + x^{3} (- 5 i) + 4 x^{3} - 8 x^{2} - 20 x^{2} i + 5 x^{3} i - 10 x^{2} i + 25 x^{2} + 30 x^{2} i - 60 x i + 150 x - 3 x^{2} + 6 x + 15 x i - 18 x - 18 \cdot - 2 - 18 (- 5 i) + 45 i x + 45 i \cdot - 2 + 45 i (- 5 i)$

Step 6.6.1.19

Multiply $- 18$ by $- 2$ .

$y = x^{4} - 2 x^{3} + x^{3} (- 5 i) + 4 x^{3} - 8 x^{2} - 20 x^{2} i + 5 x^{3} i - 10 x^{2} i + 25 x^{2} + 30 x^{2} i - 60 x i + 150 x - 3 x^{2} + 6 x + 15 x i - 18 x + 36 - 18 (- 5 i) + 45 i x + 45 i \cdot - 2 + 45 i (- 5 i)$

Step 6.6.1.20

Multiply $- 5$ by $- 18$ .

$y = x^{4} - 2 x^{3} + x^{3} (- 5 i) + 4 x^{3} - 8 x^{2} - 20 x^{2} i + 5 x^{3} i - 10 x^{2} i + 25 x^{2} + 30 x^{2} i - 60 x i + 150 x - 3 x^{2} + 6 x + 15 x i - 18 x + 36 + 90 i + 45 i x + 45 i \cdot - 2 + 45 i (- 5 i)$

Step 6.6.1.21

Multiply $- 2$ by $45$ .

$y = x^{4} - 2 x^{3} + x^{3} (- 5 i) + 4 x^{3} - 8 x^{2} - 20 x^{2} i + 5 x^{3} i - 10 x^{2} i + 25 x^{2} + 30 x^{2} i - 60 x i + 150 x - 3 x^{2} + 6 x + 15 x i - 18 x + 36 + 90 i + 45 i x - 90 i + 45 i (- 5 i)$

Step 6.6.1.22

Multiply $45 i (- 5 i)$ .

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Step 6.6.1.22.1

Multiply $- 5$ by $45$ .

$y = x^{4} - 2 x^{3} + x^{3} (- 5 i) + 4 x^{3} - 8 x^{2} - 20 x^{2} i + 5 x^{3} i - 10 x^{2} i + 25 x^{2} + 30 x^{2} i - 60 x i + 150 x - 3 x^{2} + 6 x + 15 x i - 18 x + 36 + 90 i + 45 i x - 90 i - 225 i i$

Step 6.6.1.22.2

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

$y = x^{4} - 2 x^{3} + x^{3} (- 5 i) + 4 x^{3} - 8 x^{2} - 20 x^{2} i + 5 x^{3} i - 10 x^{2} i + 25 x^{2} + 30 x^{2} i - 60 x i + 150 x - 3 x^{2} + 6 x + 15 x i - 18 x + 36 + 90 i + 45 i x - 90 i - 225 (i i)$

Step 6.6.1.22.3

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

Step 6.6.1.22.4

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

$y = x^{4} - 2 x^{3} + x^{3} (- 5 i) + 4 x^{3} - 8 x^{2} - 20 x^{2} i + 5 x^{3} i - 10 x^{2} i + 25 x^{2} + 30 x^{2} i - 60 x i + 150 x - 3 x^{2} + 6 x + 15 x i - 18 x + 36 + 90 i + 45 i x - 90 i - 225 i^{1 + 1}$

Step 6.6.1.22.5

Add $1$ and $1$ .

$y = x^{4} - 2 x^{3} + x^{3} (- 5 i) + 4 x^{3} - 8 x^{2} - 20 x^{2} i + 5 x^{3} i - 10 x^{2} i + 25 x^{2} + 30 x^{2} i - 60 x i + 150 x - 3 x^{2} + 6 x + 15 x i - 18 x + 36 + 90 i + 45 i x - 90 i - 225 i^{2}$

Step 6.6.1.23

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

$y = x^{4} - 2 x^{3} + x^{3} (- 5 i) + 4 x^{3} - 8 x^{2} - 20 x^{2} i + 5 x^{3} i - 10 x^{2} i + 25 x^{2} + 30 x^{2} i - 60 x i + 150 x - 3 x^{2} + 6 x + 15 x i - 18 x + 36 + 90 i + 45 i x - 90 i - 225 \cdot - 1$

Step 6.6.1.24

Multiply $- 225$ by $- 1$ .

$y = x^{4} - 2 x^{3} + x^{3} (- 5 i) + 4 x^{3} - 8 x^{2} - 20 x^{2} i + 5 x^{3} i - 10 x^{2} i + 25 x^{2} + 30 x^{2} i - 60 x i + 150 x - 3 x^{2} + 6 x + 15 x i - 18 x + 36 + 90 i + 45 i x - 90 i + 225$

Step 6.6.2

Simplify by adding terms.

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Step 6.6.2.1

Combine the opposite terms in $x^{4} - 2 x^{3} + x^{3} (- 5 i) + 4 x^{3} - 8 x^{2} - 20 x^{2} i + 5 x^{3} i - 10 x^{2} i + 25 x^{2} + 30 x^{2} i - 60 x i + 150 x - 3 x^{2} + 6 x + 15 x i - 18 x + 36 + 90 i + 45 i x - 90 i + 225$ .

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Step 6.6.2.1.1

Reorder the factors in the terms $x^{3} (- 5 i)$ and $5 x^{3} i$ .

$y = x^{4} - 2 x^{3} - 5 i x^{3} + 4 x^{3} - 8 x^{2} - 20 x^{2} i + 5 i x^{3} - 10 x^{2} i + 25 x^{2} + 30 x^{2} i - 60 x i + 150 x - 3 x^{2} + 6 x + 15 x i - 18 x + 36 + 90 i + 45 i x - 90 i + 225$

Step 6.6.2.1.2

Add $- 5 i x^{3}$ and $5 i x^{3}$ .

$y = x^{4} - 2 x^{3} + 4 x^{3} - 8 x^{2} - 20 x^{2} i + 0 - 10 x^{2} i + 25 x^{2} + 30 x^{2} i - 60 x i + 150 x - 3 x^{2} + 6 x + 15 x i - 18 x + 36 + 90 i + 45 i x - 90 i + 225$

Step 6.6.2.1.3

Add $x^{4} - 2 x^{3} + 4 x^{3} - 8 x^{2} - 20 x^{2} i$ and $0$ .

$y = x^{4} - 2 x^{3} + 4 x^{3} - 8 x^{2} - 20 x^{2} i - 10 x^{2} i + 25 x^{2} + 30 x^{2} i - 60 x i + 150 x - 3 x^{2} + 6 x + 15 x i - 18 x + 36 + 90 i + 45 i x - 90 i + 225$

Step 6.6.2.1.4

Subtract $90 i$ from $90 i$ .

$y = x^{4} - 2 x^{3} + 4 x^{3} - 8 x^{2} - 20 x^{2} i - 10 x^{2} i + 25 x^{2} + 30 x^{2} i - 60 x i + 150 x - 3 x^{2} + 6 x + 15 x i - 18 x + 36 + 45 i x + 0 + 225$

Step 6.6.2.1.5

Add $x^{4} - 2 x^{3} + 4 x^{3} - 8 x^{2} - 20 x^{2} i - 10 x^{2} i + 25 x^{2} + 30 x^{2} i - 60 x i + 150 x - 3 x^{2} + 6 x + 15 x i - 18 x + 36 + 45 i x$ and $0$ .

$y = x^{4} - 2 x^{3} + 4 x^{3} - 8 x^{2} - 20 x^{2} i - 10 x^{2} i + 25 x^{2} + 30 x^{2} i - 60 x i + 150 x - 3 x^{2} + 6 x + 15 x i - 18 x + 36 + 45 i x + 225$

Step 6.6.2.2

Add $- 2 x^{3}$ and $4 x^{3}$ .

$y = x^{4} + 2 x^{3} - 8 x^{2} - 20 x^{2} i - 10 x^{2} i + 25 x^{2} + 30 x^{2} i - 60 x i + 150 x - 3 x^{2} + 6 x + 15 x i - 18 x + 36 + 45 i x + 225$

Step 6.6.2.3

Add $- 8 x^{2}$ and $25 x^{2}$ .

$y = x^{4} + 2 x^{3} + 17 x^{2} - 20 x^{2} i - 10 x^{2} i + 30 x^{2} i - 60 x i + 150 x - 3 x^{2} + 6 x + 15 x i - 18 x + 36 + 45 i x + 225$

Step 6.6.2.4

Subtract $3 x^{2}$ from $17 x^{2}$ .

$y = x^{4} + 2 x^{3} + 14 x^{2} - 20 x^{2} i - 10 x^{2} i + 30 x^{2} i - 60 x i + 150 x + 6 x + 15 x i - 18 x + 36 + 45 i x + 225$

Step 6.6.2.5

Subtract $10 x^{2} i$ from $- 20 x^{2} i$ .

$y = x^{4} + 2 x^{3} + 14 x^{2} - 30 x^{2} i + 30 x^{2} i - 60 x i + 150 x + 6 x + 15 x i - 18 x + 36 + 45 i x + 225$

Step 6.6.2.6

Combine the opposite terms in $x^{4} + 2 x^{3} + 14 x^{2} - 30 x^{2} i + 30 x^{2} i - 60 x i + 150 x + 6 x + 15 x i - 18 x + 36 + 45 i x + 225$ .

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Step 6.6.2.6.1

Add $- 30 x^{2} i$ and $30 x^{2} i$ .

$y = x^{4} + 2 x^{3} + 14 x^{2} + 0 - 60 x i + 150 x + 6 x + 15 x i - 18 x + 36 + 45 i x + 225$

Step 6.6.2.6.2

Add $x^{4} + 2 x^{3} + 14 x^{2}$ and $0$ .

$y = x^{4} + 2 x^{3} + 14 x^{2} - 60 x i + 150 x + 6 x + 15 x i - 18 x + 36 + 45 i x + 225$

Step 6.6.2.7

Add $- 60 x i$ and $15 x i$ .

$y = x^{4} + 2 x^{3} + 14 x^{2} - 45 x i + 150 x + 6 x - 18 x + 36 + 45 i x + 225$

Step 6.6.2.8

Combine the opposite terms in $x^{4} + 2 x^{3} + 14 x^{2} - 45 x i + 150 x + 6 x - 18 x + 36 + 45 i x + 225$ .

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Step 6.6.2.8.1

Reorder the factors in the terms $- 45 x i$ and $45 i x$ .

$y = x^{4} + 2 x^{3} + 14 x^{2} - 45 i x + 150 x + 6 x - 18 x + 36 + 45 i x + 225$

Step 6.6.2.8.2

Add $- 45 i x$ and $45 i x$ .

$y = x^{4} + 2 x^{3} + 14 x^{2} + 150 x + 6 x - 18 x + 36 + 0 + 225$

Step 6.6.2.8.3

Add $x^{4} + 2 x^{3} + 14 x^{2} + 150 x + 6 x - 18 x + 36$ and $0$ .

$y = x^{4} + 2 x^{3} + 14 x^{2} + 150 x + 6 x - 18 x + 36 + 225$

Step 6.6.2.9

Add $150 x$ and $6 x$ .

$y = x^{4} + 2 x^{3} + 14 x^{2} + 156 x - 18 x + 36 + 225$

Step 6.6.2.10

Subtract $18 x$ from $156 x$ .

$y = x^{4} + 2 x^{3} + 14 x^{2} + 138 x + 36 + 225$

Step 6.6.2.11

Add $36$ and $225$ .

$y = x^{4} + 2 x^{3} + 14 x^{2} + 138 x + 261$

Step 7