Solve (x+2y)^3-x(x+2y)^2-2xy(x+4y)

$$( x + 2 y ) ^ { 3 } - x ( x + 2 y ) ^ { 2 } - 2 x y ( x + 4 y )$$

Evaluate

$$8y^{3}$$

Solution Steps

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(x+2y\right)^{3}.

$$x^{3}+6x^{2}y+12xy^{2}+8y^{3}-x\left(x+2y\right)^{2}-2xy\left(x+4y\right)$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+2y\right)^{2}.

$$x^{3}+6x^{2}y+12xy^{2}+8y^{3}-x\left(x^{2}+4xy+4y^{2}\right)-2xy\left(x+4y\right)$$

Use the distributive property to multiply x by x^{2}+4xy+4y^{2}.

$$x^{3}+6x^{2}y+12xy^{2}+8y^{3}-\left(x^{3}+4yx^{2}+4xy^{2}\right)-2xy\left(x+4y\right)$$

To find the opposite of x^{3}+4yx^{2}+4xy^{2}, find the opposite of each term.

$$x^{3}+6x^{2}y+12xy^{2}+8y^{3}-x^{3}-4yx^{2}-4xy^{2}-2xy\left(x+4y\right)$$

Combine x^{3} and -x^{3} to get 0.

$$6x^{2}y+12xy^{2}+8y^{3}-4yx^{2}-4xy^{2}-2xy\left(x+4y\right)$$

Combine 6x^{2}y and -4yx^{2} to get 2x^{2}y.

$$2x^{2}y+12xy^{2}+8y^{3}-4xy^{2}-2xy\left(x+4y\right)$$

Combine 12xy^{2} and -4xy^{2} to get 8xy^{2}.

$$2x^{2}y+8xy^{2}+8y^{3}-2xy\left(x+4y\right)$$

Use the distributive property to multiply -2xy by x+4y.

$$2x^{2}y+8xy^{2}+8y^{3}-2yx^{2}-8xy^{2}$$

Combine 2x^{2}y and -2yx^{2} to get 0.

$$8xy^{2}+8y^{3}-8xy^{2}$$

Combine 8xy^{2} and -8xy^{2} to get 0.

$$8y^{3}$$

Expand

$$8y^{3}$$

Solution Steps

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(x+2y\right)^{3}.

$$x^{3}+6x^{2}y+12xy^{2}+8y^{3}-x\left(x+2y\right)^{2}-2xy\left(x+4y\right)$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+2y\right)^{2}.

$$x^{3}+6x^{2}y+12xy^{2}+8y^{3}-x\left(x^{2}+4xy+4y^{2}\right)-2xy\left(x+4y\right)$$

Use the distributive property to multiply x by x^{2}+4xy+4y^{2}.

$$x^{3}+6x^{2}y+12xy^{2}+8y^{3}-\left(x^{3}+4yx^{2}+4xy^{2}\right)-2xy\left(x+4y\right)$$

To find the opposite of x^{3}+4yx^{2}+4xy^{2}, find the opposite of each term.

$$x^{3}+6x^{2}y+12xy^{2}+8y^{3}-x^{3}-4yx^{2}-4xy^{2}-2xy\left(x+4y\right)$$

Combine x^{3} and -x^{3} to get 0.

$$6x^{2}y+12xy^{2}+8y^{3}-4yx^{2}-4xy^{2}-2xy\left(x+4y\right)$$

Combine 6x^{2}y and -4yx^{2} to get 2x^{2}y.

$$2x^{2}y+12xy^{2}+8y^{3}-4xy^{2}-2xy\left(x+4y\right)$$

Combine 12xy^{2} and -4xy^{2} to get 8xy^{2}.

$$2x^{2}y+8xy^{2}+8y^{3}-2xy\left(x+4y\right)$$

Use the distributive property to multiply -2xy by x+4y.

$$2x^{2}y+8xy^{2}+8y^{3}-2yx^{2}-8xy^{2}$$

Combine 2x^{2}y and -2yx^{2} to get 0.

$$8xy^{2}+8y^{3}-8xy^{2}$$

Combine 8xy^{2} and -8xy^{2} to get 0.

$$8y^{3}$$