Basic Math Examples

$\frac{a}{a + b} - \frac{b}{b - a} - \frac{2 \cdot a \cdot b}{a^{3} - \frac{b^{3}}{1}}$

Step 1

Simplify each term.

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Step 1.1

Divide $b^{3}$ by $1$ .

$\frac{a}{a + b} - \frac{b}{b - a} - \frac{2 \cdot a \cdot b}{a^{3} - b^{3}}$

Step 1.2

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

$\frac{a}{a + b} - \frac{b}{b - a} - \frac{2 a b}{(a - b) (a^{2} + a b + b^{2})}$

Step 2

To write $\frac{a}{a + b}$ as a fraction with a common denominator, multiply by $\frac{b - a}{b - a}$ .

$\frac{a}{a + b} \cdot \frac{b - a}{b - a} - \frac{b}{b - a} - \frac{2 a b}{(a - b) (a^{2} + a b + b^{2})}$

Step 3

To write $- \frac{b}{b - a}$ as a fraction with a common denominator, multiply by $\frac{a + b}{a + b}$ .

$\frac{a}{a + b} \cdot \frac{b - a}{b - a} - \frac{b}{b - a} \cdot \frac{a + b}{a + b} - \frac{2 a b}{(a - b) (a^{2} + a b + b^{2})}$

Step 4

Write each expression with a common denominator of $(a + b) (b - a)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{a}{a + b}$ by $\frac{b - a}{b - a}$ .

$\frac{a (b - a)}{(a + b) (b - a)} - \frac{b}{b - a} \cdot \frac{a + b}{a + b} - \frac{2 a b}{(a - b) (a^{2} + a b + b^{2})}$

Step 4.2

Multiply $\frac{b}{b - a}$ by $\frac{a + b}{a + b}$ .

$\frac{a (b - a)}{(a + b) (b - a)} - \frac{b (a + b)}{(b - a) (a + b)} - \frac{2 a b}{(a - b) (a^{2} + a b + b^{2})}$

Step 4.3

Reorder the factors of $(b - a) (a + b)$ .

$\frac{a (b - a)}{(a + b) (b - a)} - \frac{b (a + b)}{(a + b) (b - a)} - \frac{2 a b}{(a - b) (a^{2} + a b + b^{2})}$

Step 5

Combine the numerators over the common denominator.

$\frac{a (b - a) - b (a + b)}{(a + b) (b - a)} - \frac{2 a b}{(a - b) (a^{2} + a b + b^{2})}$

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$\frac{a b + a (- a) - b (a + b)}{(a + b) (b - a)} - \frac{2 a b}{(a - b) (a^{2} + a b + b^{2})}$

Step 6.2

Rewrite using the commutative property of multiplication.

$\frac{a b - a \cdot a - b (a + b)}{(a + b) (b - a)} - \frac{2 a b}{(a - b) (a^{2} + a b + b^{2})}$

Step 6.3

Multiply $a$ by $a$ by adding the exponents.

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Step 6.3.1

Move $a$ .

$\frac{a b - (a \cdot a) - b (a + b)}{(a + b) (b - a)} - \frac{2 a b}{(a - b) (a^{2} + a b + b^{2})}$

Step 6.3.2

Multiply $a$ by $a$ .

$\frac{a b - a^{2} - b (a + b)}{(a + b) (b - a)} - \frac{2 a b}{(a - b) (a^{2} + a b + b^{2})}$

Step 6.4

Apply the distributive property.

$\frac{a b - a^{2} - b a - b \cdot b}{(a + b) (b - a)} - \frac{2 a b}{(a - b) (a^{2} + a b + b^{2})}$

Step 6.5

Multiply $b$ by $b$ by adding the exponents.

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Step 6.5.1

Move $b$ .

$\frac{a b - a^{2} - b a - (b \cdot b)}{(a + b) (b - a)} - \frac{2 a b}{(a - b) (a^{2} + a b + b^{2})}$

Step 6.5.2

Multiply $b$ by $b$ .

$\frac{a b - a^{2} - b a - b^{2}}{(a + b) (b - a)} - \frac{2 a b}{(a - b) (a^{2} + a b + b^{2})}$

Step 6.6

Subtract $b a$ from $a b$ .

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Step 6.6.1

Move $b$ .

$\frac{- a^{2} + a b - a b - b^{2}}{(a + b) (b - a)} - \frac{2 a b}{(a - b) (a^{2} + a b + b^{2})}$

Step 6.6.2

Subtract $a b$ from $a b$ .

$\frac{- a^{2} + 0 - b^{2}}{(a + b) (b - a)} - \frac{2 a b}{(a - b) (a^{2} + a b + b^{2})}$

Step 6.7

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

$\frac{- a^{2} - b^{2}}{(a + b) (b - a)} - \frac{2 a b}{(a - b) (a^{2} + a b + b^{2})}$

Step 7

Simplify with factoring out.

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Step 7.1

Factor $- 1$ out of $a$ .

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

Step 7.2

Factor $- 1$ out of $- b$ .

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

Step 7.3

Factor $- 1$ out of $- 1 (- a) - (b)$ .

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

Step 7.4

Reorder terms.

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

Step 8

To write $\frac{- a^{2} - b^{2}}{(a + b) (- a + b)}$ as a fraction with a common denominator, multiply by $\frac{- (a^{2} + a b + b^{2})}{- (a^{2} + a b + b^{2})}$ .

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

Step 9

To write $- \frac{2 a b}{- 1 (- a + b) (a^{2} + a b + b^{2})}$ as a fraction with a common denominator, multiply by $\frac{a + b}{a + b}$ .

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

Step 10

Write each expression with a common denominator of $(a + b) (- a + b) \cdot - 1 (a^{2} + a b + b^{2})$ , by multiplying each by an appropriate factor of $1$ .

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Step 10.1

Multiply $\frac{- a^{2} - b^{2}}{(a + b) (- a + b)}$ by $\frac{- (a^{2} + a b + b^{2})}{- (a^{2} + a b + b^{2})}$ .

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

Step 10.2

Multiply $\frac{2 a b}{- 1 (- a + b) (a^{2} + a b + b^{2})}$ by $\frac{a + b}{a + b}$ .

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

Step 10.3

Reorder the factors of $(a + b) (- a + b) (- (a^{2} + a b + b^{2}))$ .

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

Step 10.4

Reorder the factors of $- 1 (- a + b) (a^{2} + a b + b^{2}) (a + b)$ .

$\frac{(- a^{2} - b^{2}) (- (a^{2} + a b + b^{2}))}{- (- a + b) (a + b) (a^{2} + a b + b^{2})} - \frac{2 a b (a + b)}{- (- a + b) (a + b) (a^{2} + a b + b^{2})}$

Step 11

Combine the numerators over the common denominator.

$\frac{(- a^{2} - b^{2}) (- (a^{2} + a b + b^{2})) - 2 a b (a + b)}{- (- a + b) (a + b) (a^{2} + a b + b^{2})}$

Step 12

Simplify the numerator.

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Step 12.1

Apply the distributive property.

$\frac{(- a^{2} - b^{2}) (- a^{2} - (a b) - b^{2}) - 2 a b (a + b)}{- (- a + b) (a + b) (a^{2} + a b + b^{2})}$

Step 12.2

Expand $(- a^{2} - b^{2}) (- a^{2} - a b - b^{2})$ by multiplying each term in the first expression by each term in the second expression.

$\frac{- a^{2} (- a^{2}) - a^{2} (- a b) - a^{2} (- b^{2}) - b^{2} (- a^{2}) - b^{2} (- a b) - b^{2} (- b^{2}) - 2 a b (a + b)}{- (- a + b) (a + b) (a^{2} + a b + b^{2})}$

Step 12.3

Simplify each term.

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Step 12.3.1

Rewrite using the commutative property of multiplication.

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

Step 12.3.2

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

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Step 12.3.2.1

Move $a^{2}$ .

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

Step 12.3.2.2

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

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

Step 12.3.2.3

Add $2$ and $2$ .

$\frac{- 1 \cdot - 1 a^{4} - a^{2} (- a b) - a^{2} (- b^{2}) - b^{2} (- a^{2}) - b^{2} (- a b) - b^{2} (- b^{2}) - 2 a b (a + b)}{- (- a + b) (a + b) (a^{2} + a b + b^{2})}$

Step 12.3.3

Multiply $- 1$ by $- 1$ .

$\frac{1 a^{4} - a^{2} (- a b) - a^{2} (- b^{2}) - b^{2} (- a^{2}) - b^{2} (- a b) - b^{2} (- b^{2}) - 2 a b (a + b)}{- (- a + b) (a + b) (a^{2} + a b + b^{2})}$

Step 12.3.4

Multiply $a^{4}$ by $1$ .

$\frac{a^{4} - a^{2} (- a b) - a^{2} (- b^{2}) - b^{2} (- a^{2}) - b^{2} (- a b) - b^{2} (- b^{2}) - 2 a b (a + b)}{- (- a + b) (a + b) (a^{2} + a b + b^{2})}$

Step 12.3.5

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

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Step 12.3.5.1

Move $a$ .

$\frac{a^{4} - (a \cdot a^{2}) (- 1 b) - a^{2} (- b^{2}) - b^{2} (- a^{2}) - b^{2} (- a b) - b^{2} (- b^{2}) - 2 a b (a + b)}{- (- a + b) (a + b) (a^{2} + a b + b^{2})}$

Step 12.3.5.2

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

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Step 12.3.5.2.1

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

$\frac{a^{4} - (a^{1} a^{2}) (- 1 b) - a^{2} (- b^{2}) - b^{2} (- a^{2}) - b^{2} (- a b) - b^{2} (- b^{2}) - 2 a b (a + b)}{- (- a + b) (a + b) (a^{2} + a b + b^{2})}$

Step 12.3.5.2.2

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

$\frac{a^{4} - a^{1 + 2} (- 1 b) - a^{2} (- b^{2}) - b^{2} (- a^{2}) - b^{2} (- a b) - b^{2} (- b^{2}) - 2 a b (a + b)}{- (- a + b) (a + b) (a^{2} + a b + b^{2})}$

Step 12.3.5.3

Add $1$ and $2$ .

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

Step 12.3.6

Rewrite using the commutative property of multiplication.

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

Step 12.3.7

Multiply $- 1$ by $- 1$ .

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

Step 12.3.8

Multiply $a^{3}$ by $1$ .

$\frac{a^{4} + a^{3} b - a^{2} (- b^{2}) - b^{2} (- a^{2}) - b^{2} (- a b) - b^{2} (- b^{2}) - 2 a b (a + b)}{- (- a + b) (a + b) (a^{2} + a b + b^{2})}$

Step 12.3.9

Rewrite using the commutative property of multiplication.

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

Step 12.3.10

Multiply $- 1$ by $- 1$ .

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

Step 12.3.11

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

$\frac{a^{4} + a^{3} b + a^{2} b^{2} - b^{2} (- a^{2}) - b^{2} (- a b) - b^{2} (- b^{2}) - 2 a b (a + b)}{- (- a + b) (a + b) (a^{2} + a b + b^{2})}$

Step 12.3.12

Rewrite using the commutative property of multiplication.

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

Step 12.3.13

Multiply $- 1$ by $- 1$ .

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

Step 12.3.14

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

$\frac{a^{4} + a^{3} b + a^{2} b^{2} + b^{2} a^{2} - b^{2} (- a b) - b^{2} (- b^{2}) - 2 a b (a + b)}{- (- a + b) (a + b) (a^{2} + a b + b^{2})}$

Step 12.3.15

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

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Step 12.3.15.1

Move $b$ .

$\frac{a^{4} + a^{3} b + a^{2} b^{2} + b^{2} a^{2} - (b \cdot b^{2}) (- a) - b^{2} (- b^{2}) - 2 a b (a + b)}{- (- a + b) (a + b) (a^{2} + a b + b^{2})}$

Step 12.3.15.2

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

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Step 12.3.15.2.1

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

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

Step 12.3.15.2.2

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

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

Step 12.3.15.3

Add $1$ and $2$ .

$\frac{a^{4} + a^{3} b + a^{2} b^{2} + b^{2} a^{2} - b^{3} (- a) - b^{2} (- b^{2}) - 2 a b (a + b)}{- (- a + b) (a + b) (a^{2} + a b + b^{2})}$

Step 12.3.16

Rewrite using the commutative property of multiplication.

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

Step 12.3.17

Multiply $- 1$ by $- 1$ .

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

Step 12.3.18

Multiply $b^{3}$ by $1$ .

$\frac{a^{4} + a^{3} b + a^{2} b^{2} + b^{2} a^{2} + b^{3} a - b^{2} (- b^{2}) - 2 a b (a + b)}{- (- a + b) (a + b) (a^{2} + a b + b^{2})}$

Step 12.3.19

Rewrite using the commutative property of multiplication.

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

Step 12.3.20

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

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Step 12.3.20.1

Move $b^{2}$ .

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

Step 12.3.20.2

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

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

Step 12.3.20.3

Add $2$ and $2$ .

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

Step 12.3.21

Multiply $- 1$ by $- 1$ .

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

Step 12.3.22

Multiply $b^{4}$ by $1$ .

$\frac{a^{4} + a^{3} b + a^{2} b^{2} + b^{2} a^{2} + b^{3} a + b^{4} - 2 a b (a + b)}{- (- a + b) (a + b) (a^{2} + a b + b^{2})}$

Step 12.4

Add $a^{2} b^{2}$ and $b^{2} a^{2}$ .

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Step 12.4.1

Reorder $b^{2}$ and $a^{2}$ .

$\frac{a^{4} + a^{3} b + a^{2} b^{2} + a^{2} b^{2} + b^{3} a + b^{4} - 2 a b (a + b)}{- (- a + b) (a + b) (a^{2} + a b + b^{2})}$

Step 12.4.2

Add $a^{2} b^{2}$ and $a^{2} b^{2}$ .

$\frac{a^{4} + a^{3} b + 2 a^{2} b^{2} + b^{3} a + b^{4} - 2 a b (a + b)}{- (- a + b) (a + b) (a^{2} + a b + b^{2})}$

Step 12.5

Apply the distributive property.

$\frac{a^{4} + a^{3} b + 2 a^{2} b^{2} + b^{3} a + b^{4} - 2 a b a - 2 a b \cdot b}{- (- a + b) (a + b) (a^{2} + a b + b^{2})}$

Step 12.6

Multiply $a$ by $a$ by adding the exponents.

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Step 12.6.1

Move $a$ .

$\frac{a^{4} + a^{3} b + 2 a^{2} b^{2} + b^{3} a + b^{4} - 2 (a \cdot a) b - 2 a b \cdot b}{- (- a + b) (a + b) (a^{2} + a b + b^{2})}$

Step 12.6.2

Multiply $a$ by $a$ .

$\frac{a^{4} + a^{3} b + 2 a^{2} b^{2} + b^{3} a + b^{4} - 2 a^{2} b - 2 a b \cdot b}{- (- a + b) (a + b) (a^{2} + a b + b^{2})}$

Step 12.7

Multiply $b$ by $b$ by adding the exponents.

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Step 12.7.1

Move $b$ .

$\frac{a^{4} + a^{3} b + 2 a^{2} b^{2} + b^{3} a + b^{4} - 2 a^{2} b - 2 a (b \cdot b)}{- (- a + b) (a + b) (a^{2} + a b + b^{2})}$

Step 12.7.2

Multiply $b$ by $b$ .

$\frac{a^{4} + a^{3} b + 2 a^{2} b^{2} + b^{3} a + b^{4} - 2 a^{2} b - 2 a b^{2}}{- (- a + b) (a + b) (a^{2} + a b + b^{2})}$

Step 13

Simplify with factoring out.

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Step 13.1

Move the negative in front of the fraction.

$- \frac{a^{4} + a^{3} b + 2 a^{2} b^{2} + b^{3} a + b^{4} - 2 a^{2} b - 2 a b^{2}}{((- a + b) (a + b)) (a^{2} + a b + b^{2})}$

Step 13.2

Factor $- 1$ out of $- a$ .

$- \frac{a^{4} + a^{3} b + 2 a^{2} b^{2} + b^{3} a + b^{4} - 2 a^{2} b - 2 a b^{2}}{(- (a) + b) (a + b) (a^{2} + a b + b^{2})}$

Step 13.3

Factor $- 1$ out of $b$ .

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

Step 13.4

Factor $- 1$ out of $- (a) - 1 (- b)$ .

$- \frac{a^{4} + a^{3} b + 2 a^{2} b^{2} + b^{3} a + b^{4} - 2 a^{2} b - 2 a b^{2}}{- (a - b) (a + b) (a^{2} + a b + b^{2})}$

Step 13.5

Simplify the expression.

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Step 13.5.1

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

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

Step 13.5.2

Move the negative in front of the fraction.

$- - \frac{a^{4} + a^{3} b + 2 a^{2} b^{2} + b^{3} a + b^{4} - 2 a^{2} b - 2 a b^{2}}{((a - b) (a + b)) (a^{2} + a b + b^{2})}$

Step 13.5.3

Multiply $- 1$ by $- 1$ .

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

Step 13.5.4

Multiply $\frac{a^{4} + a^{3} b + 2 a^{2} b^{2} + b^{3} a + b^{4} - 2 a^{2} b - 2 a b^{2}}{((a - b) (a + b)) (a^{2} + a b + b^{2})}$ by $1$ .

$\frac{a^{4} + a^{3} b + 2 a^{2} b^{2} + b^{3} a + b^{4} - 2 a^{2} b - 2 a b^{2}}{((a - b) (a + b)) (a^{2} + a b + b^{2})}$

Step 13.5.5

Reorder factors in $\frac{a^{4} + a^{3} b + 2 a^{2} b^{2} + b^{3} a + b^{4} - 2 a^{2} b - 2 a b^{2}}{((a - b) (a + b)) (a^{2} + a b + b^{2})}$ .

$\frac{a^{4} + a^{3} b + 2 a^{2} b^{2} + b^{3} a + b^{4} - 2 a^{2} b - 2 a b^{2}}{(a - b) (a + b) (a^{2} + a b + b^{2})}$