Simplify (x^2-2y)(x^4+2x^2y+4y^2)

$(x^{2} - 2 y) (x^{4} + 2 x^{2} y + 4 y^{2})$

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

$x^{2} x^{4} + x^{2} (2 x^{2} y) + x^{2} (4 y^{2}) - 2 y x^{4} - 2 y (2 x^{2} y) - 2 y (4 y^{2})$

Simplify terms.

Simplify each term.

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

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

$x^{2 + 4} + x^{2} (2 x^{2} y) + x^{2} (4 y^{2}) - 2 y x^{4} - 2 y (2 x^{2} y) - 2 y (4 y^{2})$

Add $2$ and $4$ .

$x^{6} + x^{2} (2 x^{2} y) + x^{2} (4 y^{2}) - 2 y x^{4} - 2 y (2 x^{2} y) - 2 y (4 y^{2})$

Rewrite using the commutative property of multiplication.

$x^{6} + 2 x^{2} (x^{2} y) + x^{2} (4 y^{2}) - 2 y x^{4} - 2 y (2 x^{2} y) - 2 y (4 y^{2})$

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

Move $x^{2}$ .

$x^{6} + 2 (x^{2} x^{2}) y + x^{2} (4 y^{2}) - 2 y x^{4} - 2 y (2 x^{2} y) - 2 y (4 y^{2})$

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

$x^{6} + 2 x^{2 + 2} y + x^{2} (4 y^{2}) - 2 y x^{4} - 2 y (2 x^{2} y) - 2 y (4 y^{2})$

Add $2$ and $2$ .

$x^{6} + 2 x^{4} y + x^{2} (4 y^{2}) - 2 y x^{4} - 2 y (2 x^{2} y) - 2 y (4 y^{2})$

Rewrite using the commutative property of multiplication.

$x^{6} + 2 x^{4} y + 4 x^{2} y^{2} - 2 y x^{4} - 2 y (2 x^{2} y) - 2 y (4 y^{2})$

Multiply $y$ by $y$ by adding the exponents.

Move $y$ .

$x^{6} + 2 x^{4} y + 4 x^{2} y^{2} - 2 y x^{4} - 2 (y \cdot y) (2 x^{2}) - 2 y (4 y^{2})$

Multiply $y$ by $y$ .

$x^{6} + 2 x^{4} y + 4 x^{2} y^{2} - 2 y x^{4} - 2 y^{2} (2 x^{2}) - 2 y (4 y^{2})$

Rewrite using the commutative property of multiplication.

$x^{6} + 2 x^{4} y + 4 x^{2} y^{2} - 2 y x^{4} - 2 \cdot 2 y^{2} x^{2} - 2 y (4 y^{2})$

Multiply $- 2$ by $2$ .

$x^{6} + 2 x^{4} y + 4 x^{2} y^{2} - 2 y x^{4} - 4 y^{2} x^{2} - 2 y (4 y^{2})$

Rewrite using the commutative property of multiplication.

$x^{6} + 2 x^{4} y + 4 x^{2} y^{2} - 2 y x^{4} - 4 y^{2} x^{2} - 2 \cdot 4 y \cdot y^{2}$

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

Move $y^{2}$ .

$x^{6} + 2 x^{4} y + 4 x^{2} y^{2} - 2 y x^{4} - 4 y^{2} x^{2} - 2 \cdot 4 (y^{2} y)$

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

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

$x^{6} + 2 x^{4} y + 4 x^{2} y^{2} - 2 y x^{4} - 4 y^{2} x^{2} - 2 \cdot 4 (y^{2} y^{1})$

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

$x^{6} + 2 x^{4} y + 4 x^{2} y^{2} - 2 y x^{4} - 4 y^{2} x^{2} - 2 \cdot 4 y^{2 + 1}$

Add $2$ and $1$ .

$x^{6} + 2 x^{4} y + 4 x^{2} y^{2} - 2 y x^{4} - 4 y^{2} x^{2} - 2 \cdot 4 y^{3}$

Multiply $- 2$ by $4$ .

$x^{6} + 2 x^{4} y + 4 x^{2} y^{2} - 2 y x^{4} - 4 y^{2} x^{2} - 8 y^{3}$

Combine the opposite terms in $x^{6} + 2 x^{4} y + 4 x^{2} y^{2} - 2 y x^{4} - 4 y^{2} x^{2} - 8 y^{3}$ .

Reorder the factors in the terms $2 x^{4} y$ and $- 2 y x^{4}$ .

$x^{6} + 2 x^{4} y + 4 x^{2} y^{2} - 2 x^{4} y - 4 y^{2} x^{2} - 8 y^{3}$

Subtract $2 x^{4} y$ from $2 x^{4} y$ .

$x^{6} + 4 x^{2} y^{2} + 0 - 4 y^{2} x^{2} - 8 y^{3}$

Add $x^{6} + 4 x^{2} y^{2}$ and $0$ .

$x^{6} + 4 x^{2} y^{2} - 4 y^{2} x^{2} - 8 y^{3}$

Reorder the factors in the terms $4 x^{2} y^{2}$ and $- 4 y^{2} x^{2}$ .

$x^{6} + 4 x^{2} y^{2} - 4 x^{2} y^{2} - 8 y^{3}$

Subtract $4 x^{2} y^{2}$ from $4 x^{2} y^{2}$ .

$x^{6} + 0 - 8 y^{3}$

Add $x^{6}$ and $0$ .

$x^{6} - 8 y^{3}$

Simplify (x^2-2y)(x^4+2x^2y+4y^2)

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