Factor (m+n)^4(m-n)-(m-n)^4(m+n)

${(m + n)}^{4} (m - n) - {(m - n)}^{4} (m + n)$

Factor $(m + n) (m - n)$ out of ${(m + n)}^{4} (m - n) - {(m - n)}^{4} (m + n)$ .

Factor $(m + n) (m - n)$ out of ${(m + n)}^{4} (m - n)$ .

$(m + n) (m - n) ({(m + n)}^{3}) - {(m - n)}^{4} (m + n)$

Factor $(m + n) (m - n)$ out of $- {(m - n)}^{4} (m + n)$ .

$(m + n) (m - n) ({(m + n)}^{3}) + (m + n) (m - n) (- {(m - n)}^{3})$

Factor $(m + n) (m - n)$ out of $(m + n) (m - n) ({(m + n)}^{3}) + (m + n) (m - n) (- {(m - n)}^{3})$ .

$(m + n) (m - n) ({(m + n)}^{3} - {(m - n)}^{3})$

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 $a = m + n$ and $b = m - n$ .

$(m + n) (m - n) ((m + n - (m - n)) ({(m + n)}^{2} + (m + n) (m - n) + {(m - n)}^{2}))$

Factor.

Simplify.

Apply the distributive property.

$(m + n) (m - n) ((m + n - m - - n) ({(m + n)}^{2} + (m + n) (m - n) + {(m - n)}^{2}))$

Multiply $- - n$ .

Multiply $- 1$ by $- 1$ .

$(m + n) (m - n) ((m + n - m + 1 n) ({(m + n)}^{2} + (m + n) (m - n) + {(m - n)}^{2}))$

Multiply $n$ by $1$ .

$(m + n) (m - n) ((m + n - m + n) ({(m + n)}^{2} + (m + n) (m - n) + {(m - n)}^{2}))$

Subtract $m$ from $m$ .

$(m + n) (m - n) ((n + 0 + n) ({(m + n)}^{2} + (m + n) (m - n) + {(m - n)}^{2}))$

Add $n$ and $0$ .

$(m + n) (m - n) ((n + n) ({(m + n)}^{2} + (m + n) (m - n) + {(m - n)}^{2}))$

Add $n$ and $n$ .

$(m + n) (m - n) (2 n ({(m + n)}^{2} + (m + n) (m - n) + {(m - n)}^{2}))$

Rewrite ${(m + n)}^{2}$ as $(m + n) (m + n)$ .

$(m + n) (m - n) (2 n ((m + n) (m + n) + (m + n) (m - n) + {(m - n)}^{2}))$

Expand $(m + n) (m + n)$ using the FOIL Method.

Apply the distributive property.

$(m + n) (m - n) (2 n (m (m + n) + n (m + n) + (m + n) (m - n) + {(m - n)}^{2}))$

Apply the distributive property.

$(m + n) (m - n) (2 n (m \cdot m + m n + n (m + n) + (m + n) (m - n) + {(m - n)}^{2}))$

Apply the distributive property.

$(m + n) (m - n) (2 n (m \cdot m + m n + n m + n \cdot n + (m + n) (m - n) + {(m - n)}^{2}))$

Simplify and combine like terms.

Simplify each term.

Multiply $m$ by $m$ .

$(m + n) (m - n) (2 n (m^{2} + m n + n m + n \cdot n + (m + n) (m - n) + {(m - n)}^{2}))$

Multiply $n$ by $n$ .

$(m + n) (m - n) (2 n (m^{2} + m n + n m + n^{2} + (m + n) (m - n) + {(m - n)}^{2}))$

Add $m n$ and $n m$ .

Reorder $n$ and $m$ .

$(m + n) (m - n) (2 n (m^{2} + m n + m n + n^{2} + (m + n) (m - n) + {(m - n)}^{2}))$

Add $m n$ and $m n$ .

$(m + n) (m - n) (2 n (m^{2} + 2 m n + n^{2} + (m + n) (m - n) + {(m - n)}^{2}))$

Expand $(m + n) (m - n)$ using the FOIL Method.

Apply the distributive property.

$(m + n) (m - n) (2 n (m^{2} + 2 m n + n^{2} + m (m - n) + n (m - n) + {(m - n)}^{2}))$

Apply the distributive property.

$(m + n) (m - n) (2 n (m^{2} + 2 m n + n^{2} + m \cdot m + m (- n) + n (m - n) + {(m - n)}^{2}))$

Apply the distributive property.

$(m + n) (m - n) (2 n (m^{2} + 2 m n + n^{2} + m \cdot m + m (- n) + n m + n (- n) + {(m - n)}^{2}))$

Combine the opposite terms in $m \cdot m + m (- n) + n m + n (- n)$ .

Reorder the factors in the terms $m (- n)$ and $n m$ .

$(m + n) (m - n) (2 n (m^{2} + 2 m n + n^{2} + m \cdot m - m n + m n + n (- n) + {(m - n)}^{2}))$

Add $- m n$ and $m n$ .

$(m + n) (m - n) (2 n (m^{2} + 2 m n + n^{2} + m \cdot m + 0 + n (- n) + {(m - n)}^{2}))$

Add $m \cdot m$ and $0$ .

$(m + n) (m - n) (2 n (m^{2} + 2 m n + n^{2} + m \cdot m + n (- n) + {(m - n)}^{2}))$

Simplify each term.

Multiply $m$ by $m$ .

$(m + n) (m - n) (2 n (m^{2} + 2 m n + n^{2} + m^{2} + n (- n) + {(m - n)}^{2}))$

Rewrite using the commutative property of multiplication.

$(m + n) (m - n) (2 n (m^{2} + 2 m n + n^{2} + m^{2} - n \cdot n + {(m - n)}^{2}))$

Multiply $n$ by $n$ by adding the exponents.

Move $n$ .

$(m + n) (m - n) (2 n (m^{2} + 2 m n + n^{2} + m^{2} - (n \cdot n) + {(m - n)}^{2}))$

Multiply $n$ by $n$ .

$(m + n) (m - n) (2 n (m^{2} + 2 m n + n^{2} + m^{2} - n^{2} + {(m - n)}^{2}))$

Rewrite ${(m - n)}^{2}$ as $(m - n) (m - n)$ .

$(m + n) (m - n) (2 n (m^{2} + 2 m n + n^{2} + m^{2} - n^{2} + (m - n) (m - n)))$

Expand $(m - n) (m - n)$ using the FOIL Method.

Apply the distributive property.

$(m + n) (m - n) (2 n (m^{2} + 2 m n + n^{2} + m^{2} - n^{2} + m (m - n) - n (m - n)))$

Apply the distributive property.

$(m + n) (m - n) (2 n (m^{2} + 2 m n + n^{2} + m^{2} - n^{2} + m \cdot m + m (- n) - n (m - n)))$

Apply the distributive property.

$(m + n) (m - n) (2 n (m^{2} + 2 m n + n^{2} + m^{2} - n^{2} + m \cdot m + m (- n) - n m - n (- n)))$

Simplify and combine like terms.

Simplify each term.

Multiply $m$ by $m$ .

$(m + n) (m - n) (2 n (m^{2} + 2 m n + n^{2} + m^{2} - n^{2} + m^{2} + m (- n) - n m - n (- n)))$

Rewrite using the commutative property of multiplication.

$(m + n) (m - n) (2 n (m^{2} + 2 m n + n^{2} + m^{2} - n^{2} + m^{2} - m n - n m - n (- n)))$

Rewrite using the commutative property of multiplication.

$(m + n) (m - n) (2 n (m^{2} + 2 m n + n^{2} + m^{2} - n^{2} + m^{2} - m n - n m - 1 \cdot - 1 n \cdot n))$

Multiply $n$ by $n$ by adding the exponents.

Move $n$ .

$(m + n) (m - n) (2 n (m^{2} + 2 m n + n^{2} + m^{2} - n^{2} + m^{2} - m n - n m - 1 \cdot - 1 (n \cdot n)))$

Multiply $n$ by $n$ .

$(m + n) (m - n) (2 n (m^{2} + 2 m n + n^{2} + m^{2} - n^{2} + m^{2} - m n - n m - 1 \cdot - 1 n^{2}))$

Multiply $- 1$ by $- 1$ .

$(m + n) (m - n) (2 n (m^{2} + 2 m n + n^{2} + m^{2} - n^{2} + m^{2} - m n - n m + 1 n^{2}))$

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

$(m + n) (m - n) (2 n (m^{2} + 2 m n + n^{2} + m^{2} - n^{2} + m^{2} - m n - n m + n^{2}))$

Subtract $n m$ from $- m n$ .

Move $n$ .

$(m + n) (m - n) (2 n (m^{2} + 2 m n + n^{2} + m^{2} - n^{2} + m^{2} - m n - 1 m n + n^{2}))$

Subtract $m n$ from $- m n$ .

$(m + n) (m - n) (2 n (m^{2} + 2 m n + n^{2} + m^{2} - n^{2} + m^{2} - 2 m n + n^{2}))$

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

$(m + n) (m - n) (2 n (2 m^{2} + 2 m n + n^{2} - n^{2} + m^{2} - 2 m n + n^{2}))$

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

$(m + n) (m - n) (2 n (3 m^{2} + 2 m n + n^{2} - n^{2} - 2 m n + n^{2}))$

Subtract $2 m n$ from $2 m n$ .

$(m + n) (m - n) (2 n (3 m^{2} + n^{2} - n^{2} + 0 + n^{2}))$

Add $3 m^{2}$ and $0$ .

$(m + n) (m - n) (2 n (3 m^{2} + n^{2} - n^{2} + n^{2}))$

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

$(m + n) (m - n) (2 n (3 m^{2} + 0 + n^{2}))$

Add $3 m^{2}$ and $0$ .

$(m + n) (m - n) (2 n (3 m^{2} + n^{2}))$

Remove unnecessary parentheses.

$(m + n) (m - n) \cdot 2 n (3 m^{2} + n^{2})$

Factor (m+n)^4(m-n)-(m-n)^4(m+n)

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