Simplify (p^2+q^2)(p^4-pq-q^4)

$(p^{2} + q^{2}) (p^{4} - p q - q^{4})$

Expand $(p^{2} + q^{2}) (p^{4} - p q - q^{4})$ by multiplying each term in the first expression by each term in the second expression.

$p^{2} p^{4} + p^{2} (- p q) + p^{2} (- q^{4}) + q^{2} p^{4} + q^{2} (- p q) + q^{2} (- q^{4})$

Simplify terms.

Simplify each term.

Multiply $p^{2}$ by $p^{4}$ by adding the exponents.

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

$p^{2 + 4} + p^{2} (- p q) + p^{2} (- q^{4}) + q^{2} p^{4} + q^{2} (- p q) + q^{2} (- q^{4})$

Add $2$ and $4$ .

$p^{6} + p^{2} (- p q) + p^{2} (- q^{4}) + q^{2} p^{4} + q^{2} (- p q) + q^{2} (- q^{4})$

Rewrite using the commutative property of multiplication.

$p^{6} - p^{2} (p q) + p^{2} (- q^{4}) + q^{2} p^{4} + q^{2} (- p q) + q^{2} (- q^{4})$

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

Move $p$ .

$p^{6} - (p \cdot p^{2}) q + p^{2} (- q^{4}) + q^{2} p^{4} + q^{2} (- p q) + q^{2} (- q^{4})$

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

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

$p^{6} - (p^{1} p^{2}) q + p^{2} (- q^{4}) + q^{2} p^{4} + q^{2} (- p q) + q^{2} (- q^{4})$

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

$p^{6} - p^{1 + 2} q + p^{2} (- q^{4}) + q^{2} p^{4} + q^{2} (- p q) + q^{2} (- q^{4})$

Add $1$ and $2$ .

$p^{6} - p^{3} q + p^{2} (- q^{4}) + q^{2} p^{4} + q^{2} (- p q) + q^{2} (- q^{4})$

Rewrite using the commutative property of multiplication.

$p^{6} - p^{3} q - p^{2} q^{4} + q^{2} p^{4} + q^{2} (- p q) + q^{2} (- q^{4})$

Rewrite using the commutative property of multiplication.

$p^{6} - p^{3} q - p^{2} q^{4} + q^{2} p^{4} - q^{2} (p q) + q^{2} (- q^{4})$

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

Move $q$ .

$p^{6} - p^{3} q - p^{2} q^{4} + q^{2} p^{4} - (q \cdot q^{2}) p + q^{2} (- q^{4})$

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

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

$p^{6} - p^{3} q - p^{2} q^{4} + q^{2} p^{4} - (q^{1} q^{2}) p + q^{2} (- q^{4})$

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

$p^{6} - p^{3} q - p^{2} q^{4} + q^{2} p^{4} - q^{1 + 2} p + q^{2} (- q^{4})$

Add $1$ and $2$ .

$p^{6} - p^{3} q - p^{2} q^{4} + q^{2} p^{4} - q^{3} p + q^{2} (- q^{4})$

Rewrite using the commutative property of multiplication.

$p^{6} - p^{3} q - p^{2} q^{4} + q^{2} p^{4} - q^{3} p - q^{2} q^{4}$

Multiply $q^{2}$ by $q^{4}$ by adding the exponents.

Move $q^{4}$ .

$p^{6} - p^{3} q - p^{2} q^{4} + q^{2} p^{4} - q^{3} p - (q^{4} q^{2})$

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

$p^{6} - p^{3} q - p^{2} q^{4} + q^{2} p^{4} - q^{3} p - q^{4 + 2}$

Add $4$ and $2$ .

$p^{6} - p^{3} q - p^{2} q^{4} + q^{2} p^{4} - q^{3} p - q^{6}$

Simplify the expression.

Reorder $q^{2}$ and $p^{4}$ .

$p^{6} - p^{3} q - p^{2} q^{4} + p^{4} q^{2} - q^{3} p - q^{6}$

Move $q^{3}$ .

$p^{6} - p^{3} q - p^{2} q^{4} + p^{4} q^{2} - p q^{3} - q^{6}$

Move $- p q^{3}$ .

$p^{6} - p^{3} q - p^{2} q^{4} + p^{4} q^{2} - q^{6} - p q^{3}$

Move $- p^{3} q$ .

$p^{6} - p^{2} q^{4} + p^{4} q^{2} - q^{6} - p^{3} q - p q^{3}$

Move $- p^{2} q^{4}$ .

$p^{6} + p^{4} q^{2} - p^{2} q^{4} - q^{6} - p^{3} q - p q^{3}$

Simplify (p^2+q^2)(p^4-pq-q^4)

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