Factor (a+b)^4-(a-b)^4

${(a + b)}^{4} - {(a - b)}^{4}$

Rewrite ${(a + b)}^{4}$ as ${({(a + b)}^{2})}^{2}$ .

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

Rewrite ${(a - b)}^{4}$ as ${({(a - b)}^{2})}^{2}$ .

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

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

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

Simplify.

Rewrite ${(a + b)}^{2}$ as $(a + b) (a + b)$ .

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

Expand $(a + b) (a + b)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $a$ by $a$ .

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

Multiply $b$ by $b$ .

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

Add $a b$ and $b a$ .

Reorder $b$ and $a$ .

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

Add $a b$ and $a b$ .

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

Rewrite ${(a - b)}^{2}$ as $(a - b) (a - b)$ .

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

Expand $(a - b) (a - b)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $a$ by $a$ .

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

Rewrite using the commutative property of multiplication.

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

Rewrite using the commutative property of multiplication.

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

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

Move $b$ .

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

Multiply $b$ by $b$ .

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

Multiply $- 1$ by $- 1$ .

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

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

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

Subtract $b a$ from $- a b$ .

Move $b$ .

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

Subtract $a b$ from $- a b$ .

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

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

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

Subtract $2 a b$ from $2 a b$ .

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

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

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

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

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

Factor $2$ out of $2 a^{2} + 2 b^{2}$ .

Factor $2$ out of $2 a^{2}$ .

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

Factor $2$ out of $2 b^{2}$ .

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

Factor $2$ out of $2 (a^{2}) + 2 (b^{2})$ .

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

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

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

Factor.

Simplify.

Remove unnecessary parentheses.

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

Add $a$ and $a$ .

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

Subtract $b$ from $b$ .

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

Add $2 a$ and $0$ .

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

Apply the distributive property.

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

Multiply $- - b$ .

Multiply $- 1$ by $- 1$ .

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

Multiply $b$ by $1$ .

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

Subtract $a$ from $a$ .

$2 (a^{2} + b^{2}) (2 a (b + 0 + b))$

Add $b$ and $0$ .

$2 (a^{2} + b^{2}) (2 a (b + b))$

Factor.

Add $b$ and $b$ .

$2 (a^{2} + b^{2}) (2 a (2 b))$

Remove unnecessary parentheses.

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

Multiply $2$ by $2$ .

$2 (a^{2} + b^{2}) (4 a b)$

Remove unnecessary parentheses.

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

Multiply $4$ by $2$ .

$8 (a^{2} + b^{2}) a b$

Factor (a+b)^4-(a-b)^4

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