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

$(x - y) (x + y) (2 x + y)$

Step 1

Expand $(x - y) (x + y)$ using the FOIL Method.

Apply the distributive property.

$(x (x + y) - y (x + y)) (2 x + y)$

Apply the distributive property.

$(x \cdot x + x y - y (x + y)) (2 x + y)$

Apply the distributive property.

$(x \cdot x + x y - y x - y \cdot y) (2 x + y)$

Step 2

Simplify terms.

Combine the opposite terms in $x \cdot x + x y - y x - y \cdot y$ .

Reorder the factors in the terms $x y$ and $- y x$ .

$(x \cdot x + x y - x y - y \cdot y) (2 x + y)$

Subtract $x y$ from $x y$ .

$(x \cdot x + 0 - y \cdot y) (2 x + y)$

Add $x \cdot x$ and $0$ .

$(x \cdot x - y \cdot y) (2 x + y)$

Simplify each term.

Multiply $x$ by $x$ .

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

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

Move $y$ .

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

Multiply $y$ by $y$ .

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

Step 3

Expand $(x^{2} - y^{2}) (2 x + y)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 4

Simplify terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

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

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

Move $x$ .

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

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

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

$2 (x^{1} x^{2}) + x^{2} y - y^{2} (2 x) - y^{2} y$

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

$2 x^{1 + 2} + x^{2} y - y^{2} (2 x) - y^{2} y$

Add $1$ and $2$ .

$2 x^{3} + x^{2} y - y^{2} (2 x) - y^{2} y$

Rewrite using the commutative property of multiplication.

$2 x^{3} + x^{2} y - 1 \cdot 2 y^{2} x - y^{2} y$

Multiply $- 1$ by $2$ .

$2 x^{3} + x^{2} y - 2 y^{2} x - y^{2} y$

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

Move $y$ .

$2 x^{3} + x^{2} y - 2 y^{2} x - (y \cdot y^{2})$

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

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

$2 x^{3} + x^{2} y - 2 y^{2} x - (y^{1} y^{2})$

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

$2 x^{3} + x^{2} y - 2 y^{2} x - y^{1 + 2}$

Add $1$ and $2$ .

$2 x^{3} + x^{2} y - 2 y^{2} x - y^{3}$

Move $y^{2}$ .

$2 x^{3} + x^{2} y - 2 x y^{2} - y^{3}$

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

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