Precalculus Examples

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

Step 1

Find where the expression $\frac{- 2 x^{4} + x^{3} - 2 x + 1}{x^{3} + 3 x^{2}}$ is undefined.

$x = - 3, x = 0$

Step 2

Since $\frac{- 2 x^{4} + x^{3} - 2 x + 1}{x^{3} + 3 x^{2}}$ $\to$ $\infty$ as $x$ $\to$ $- 3$ from the left and $\frac{- 2 x^{4} + x^{3} - 2 x + 1}{x^{3} + 3 x^{2}}$ $\to$ $- \infty$ as $x$ $\to$ $- 3$ from the right, then $x = - 3$ is a vertical asymptote.

$x = - 3$

Step 3

Since $\frac{- 2 x^{4} + x^{3} - 2 x + 1}{x^{3} + 3 x^{2}}$ $\to$ $\infty$ as $x$ $\to$ $0$ from the left and $\frac{- 2 x^{4} + x^{3} - 2 x + 1}{x^{3} + 3 x^{2}}$ $\to$ $\infty$ as $x$ $\to$ $0$ from the right, then $x = 0$ is a vertical asymptote.

$x = 0$

Step 4

List all of the vertical asymptotes:

$x = - 3, 0$

Step 5

Consider the rational function $R (x) = \frac{a x^{n}}{b x^{m}}$ where $n$ is the degree of the numerator and $m$ is the degree of the denominator.

1. If $n < m$ , then the x-axis, $y = 0$ , is the horizontal asymptote.

2. If $n = m$ , then the horizontal asymptote is the line $y = \frac{a}{b}$ .

3. If $n > m$ , then there is no horizontal asymptote (there is an oblique asymptote).

Step 6

Find $n$ and $m$ .

$n = 4$

$m = 3$

Step 7

Since $n > m$ , there is no horizontal asymptote.

No Horizontal Asymptotes

Step 8

Find the oblique asymptote using polynomial division.

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Step 8.1

Simplify the expression.

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Step 8.1.1

Simplify the numerator.

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Step 8.1.1.1

Factor out the greatest common factor from each group.

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Step 8.1.1.1.1

Group the first two terms and the last two terms.

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

Step 8.1.1.1.2

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

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

Step 8.1.1.2

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

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

Step 8.1.1.3

Rewrite $1$ as $1^{3}$ .

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

Step 8.1.1.4

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = x$ and $b = 1$ .

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

Step 8.1.1.5

Simplify.

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Step 8.1.1.5.1

Multiply $- 1$ by $1$ .

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

Step 8.1.1.5.2

One to any power is one.

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

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

Step 8.1.2

Simplify with factoring out.

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Step 8.1.2.1

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

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Step 8.1.2.1.1

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

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

Step 8.1.2.1.2

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

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

Step 8.1.2.1.3

Factor $x^{2}$ out of $x^{2} x + x^{2} \cdot 3$ .

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

Step 8.1.2.2

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

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

Step 8.1.2.3

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

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

Step 8.1.2.4

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

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

Step 8.1.2.5

Simplify the expression.

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Step 8.1.2.5.1

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

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

Step 8.1.2.5.2

Move the negative in front of the fraction.

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

Step 8.1.2.5.3

Reorder factors in $- \frac{((2 x - 1) (x + 1)) (x^{2} - x + 1)}{x^{2} (x + 3)}$ .

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

Step 8.2

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

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Step 8.2.1

Negate $2 x - 1$ .

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

Step 8.2.2

Apply the distributive property.

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

Step 8.2.3

Apply the distributive property.

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

Step 8.2.4

Apply the distributive property.

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

Step 8.2.5

Apply the distributive property.

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

Step 8.2.6

Apply the distributive property.

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

Step 8.2.7

Apply the distributive property.

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

Step 8.2.8

Apply the distributive property.

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

Step 8.2.9

Apply the distributive property.

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

Step 8.2.10

Apply the distributive property.

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

Step 8.2.11

Apply the distributive property.

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

Step 8.2.12

Apply the distributive property.

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

Step 8.2.13

Apply the distributive property.

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

Step 8.2.14

Apply the distributive property.

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

Step 8.2.15

Apply the distributive property.

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

Step 8.2.16

Apply the distributive property.

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

Step 8.2.17

Remove parentheses.

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

Step 8.2.18

Remove parentheses.

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

Step 8.2.19

Remove parentheses.

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

Step 8.2.20

Move $x$ .

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

Step 8.2.21

Move $x$ .

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

Step 8.2.22

Remove parentheses.

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

Step 8.2.23

Move $x$ .

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

Step 8.2.24

Move $x$ .

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

Step 8.2.25

Remove parentheses.

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

Step 8.2.26

Move $x$ .

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

Step 8.2.27

Remove parentheses.

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

Step 8.2.28

Move $x$ .

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

Step 8.2.29

Remove parentheses.

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

Step 8.2.30

Move $x$ .

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

Step 8.2.31

Remove parentheses.

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

Step 8.2.32

Move $x$ .

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

Step 8.2.33

Move $x$ .

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

Step 8.2.34

Remove parentheses.

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

Step 8.2.35

Move $x$ .

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

Step 8.2.36

Move $x$ .

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

Step 8.2.37

Remove parentheses.

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

Step 8.2.38

Multiply $- 1$ by $2$ .

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

Step 8.2.39

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

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

Step 8.2.40

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

Step 8.2.41

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

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

Step 8.2.42

Add $1$ and $1$ .

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

Step 8.2.43

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

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

Step 8.2.44

Add $2$ and $2$ .

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

Step 8.2.45

Multiply $- 1$ by $2$ .

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

Step 8.2.46

Multiply $- 2$ by $- 1$ .

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

Step 8.2.47

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

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

Step 8.2.48

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

Step 8.2.49

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

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

Step 8.2.50

Add $1$ and $1$ .

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

Step 8.2.51

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

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

Step 8.2.52

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

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

Step 8.2.53

Add $2$ and $1$ .

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

Step 8.2.54

Multiply $- 1$ by $2$ .

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

Step 8.2.55

Multiply $- 2$ by $1$ .

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

Step 8.2.56

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

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

Step 8.2.57

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

Step 8.2.58

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

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

Step 8.2.59

Add $1$ and $1$ .

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

Step 8.2.60

Multiply $- 1$ by $2$ .

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

Step 8.2.61

Multiply $- 2$ by $1$ .

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

Step 8.2.62

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

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

Step 8.2.63

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

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

Step 8.2.64

Add $1$ and $2$ .

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

Step 8.2.65

Multiply $- 1$ by $2$ .

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

Step 8.2.66

Multiply $- 2$ by $1$ .

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

Step 8.2.67

Multiply $- 2$ by $- 1$ .

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

Step 8.2.68

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

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

Step 8.2.69

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

Step 8.2.70

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

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

Step 8.2.71

Add $1$ and $1$ .

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

Step 8.2.72

Multiply $- 1$ by $2$ .

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

Step 8.2.73

Multiply $- 2$ by $1$ .

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

Step 8.2.74

Multiply $- 2$ by $1$ .

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

Step 8.2.75

Multiply $- 1$ by $- 1$ .

Step 8.2.76

Multiply $x$ by $1$ .

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

Step 8.2.77

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

Step 8.2.78

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

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

Step 8.2.79

Add $1$ and $2$ .

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

Step 8.2.80

Multiply $- 1$ by $- 1$ .

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

Step 8.2.81

Multiply $- 1$ by $1$ .

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

Step 8.2.82

Factor out negative.

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

Step 8.2.83

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

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

Step 8.2.84

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

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

Step 8.2.85

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

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

Step 8.2.86

Add $1$ and $1$ .

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

Step 8.2.87

Multiply $- 1$ by $- 1$ .

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

Step 8.2.88

Multiply $1$ by $1$ .

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

Step 8.2.89

Multiply $x$ by $1$ .

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

Step 8.2.90

Multiply $- 1$ by $- 1$ .

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

Step 8.2.91

Multiply $1$ by $1$ .

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

Step 8.2.92

Multiply $x^{2}$ by $1$ .

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

Step 8.2.93

Multiply $- 1$ by $- 1$ .

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

Step 8.2.94

Multiply $1$ by $1$ .

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

Step 8.2.95

Multiply $- 1$ by $1$ .

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

Step 8.2.96

Multiply $- 1$ by $- 1$ .

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

Step 8.2.97

Multiply $1$ by $1$ .

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

Step 8.2.98

Multiply $1$ by $1$ .

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

Step 8.2.99

Move $- 2 x^{2}$ .

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

Step 8.2.100

Move $x$ .

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

Step 8.2.101

Move $- x^{2}$ .

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

Step 8.2.102

Move $- 2 x$ .

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

Step 8.2.103

Move $- 2 x^{2}$ .

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

Step 8.2.104

Move $2 x^{2}$ .

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

Step 8.2.105

Move $- 2 x^{3}$ .

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

Step 8.2.106

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

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

Step 8.2.107

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

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

Step 8.2.108

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

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

Step 8.2.109

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

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

Step 8.2.110

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

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

Step 8.2.111

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

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

Step 8.2.112

Subtract $x$ from $x$ .

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

Step 8.2.113

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

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

Step 8.3

Expand $x^{2} (x + 3)$ .

Tap for more steps...

Step 8.3.1

Apply the distributive property.

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

Step 8.3.2

Reorder $x^{2}$ and $3$ .

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

Step 8.3.3

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

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

Step 8.3.4

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

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

Step 8.3.5

Add $2$ and $1$ .

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

Step 8.4

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

$x^{3}$

$3 x^{2}$

$0 x$

$0$

$2 x^{4}$

$x^{3}$

$0 x^{2}$

$2 x$

$1$

Step 8.5

Divide the highest order term in the dividend $- 2 x^{4}$ by the highest order term in divisor $x^{3}$ .

$2 x$

$x^{3}$

$3 x^{2}$

$0 x$

$0$

$2 x^{4}$

$x^{3}$

$0 x^{2}$

$2 x$

$1$

Step 8.6

Multiply the new quotient term by the divisor.

$2 x$

$x^{3}$

$3 x^{2}$

$0 x$

$0$

$2 x^{4}$

$x^{3}$

$0 x^{2}$

$2 x$

$1$

$2 x^{4}$

$6 x^{3}$

$0$

Step 8.7

The expression needs to be subtracted from the dividend, so change all the signs in $- 2 x^{4} - 6 x^{3} + 0 + 0$

$2 x$

$x^{3}$

$3 x^{2}$

$0 x$

$0$

$2 x^{4}$

$x^{3}$

$0 x^{2}$

$2 x$

$1$

$2 x^{4}$

$6 x^{3}$

$0$

Step 8.8

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

$2 x$

$x^{3}$

$3 x^{2}$

$0 x$

$0$

$2 x^{4}$

$x^{3}$

$0 x^{2}$

$2 x$

$1$

$2 x^{4}$

$6 x^{3}$

$0$

$7 x^{3}$

$0$

$2 x$

Step 8.9

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

$2 x$

$x^{3}$

$3 x^{2}$

$0 x$

$0$

$2 x^{4}$

$x^{3}$

$0 x^{2}$

$2 x$

$1$

$2 x^{4}$

$6 x^{3}$

$0$

$7 x^{3}$

$0$

$2 x$

$1$

Step 8.10

Divide the highest order term in the dividend $7 x^{3}$ by the highest order term in divisor $x^{3}$ .

$2 x$

$7$

$x^{3}$

$3 x^{2}$

$0 x$

$0$

$2 x^{4}$

$x^{3}$

$0 x^{2}$

$2 x$

$1$

$2 x^{4}$

$6 x^{3}$

$0$

$7 x^{3}$

$0$

$2 x$

$1$

Step 8.11

Multiply the new quotient term by the divisor.

$2 x$

$7$

$x^{3}$

$3 x^{2}$

$0 x$

$0$

$2 x^{4}$

$x^{3}$

$0 x^{2}$

$2 x$

$1$

$2 x^{4}$

$6 x^{3}$

$0$

$7 x^{3}$

$0$

$2 x$

$1$

$7 x^{3}$

$21 x^{2}$

$0$

Step 8.12

The expression needs to be subtracted from the dividend, so change all the signs in $7 x^{3} + 21 x^{2} + 0 + 0$

							-	$2 x$	+	$7$
$x^{3}$	+	$3 x^{2}$	+	$0 x$	+	$0$	-	$2 x^{4}$	+	$x^{3}$	+	$0 x^{2}$	-	$2 x$	+	$1$
							+	$2 x^{4}$	+	$6 x^{3}$	-	$0$	-	$0$
									+	$7 x^{3}$	+	$0$	-	$2 x$	+	$1$
									-	$7 x^{3}$	-	$21 x^{2}$	-	$0$	-	$0$

Step 8.13

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

							-	$2 x$	+	$7$
$x^{3}$	+	$3 x^{2}$	+	$0 x$	+	$0$	-	$2 x^{4}$	+	$x^{3}$	+	$0 x^{2}$	-	$2 x$	+	$1$
							+	$2 x^{4}$	+	$6 x^{3}$	-	$0$	-	$0$
									+	$7 x^{3}$	+	$0$	-	$2 x$	+	$1$
									-	$7 x^{3}$	-	$21 x^{2}$	-	$0$	-	$0$
											-	$21 x^{2}$	-	$2 x$	+	$1$

Step 8.14

The final answer is the quotient plus the remainder over the divisor.

$- 2 x + 7 + \frac{- 21 x^{2} - 2 x + 1}{x^{3} + 3 x^{2}}$

Step 8.15

The oblique asymptote is the polynomial portion of the long division result.

$y = - 2 x + 7$

Step 9

This is the set of all asymptotes.

Vertical Asymptotes: $x = - 3, 0$

No Horizontal Asymptotes

Oblique Asymptotes: $y = - 2 x + 7$

Step 10