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

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

Evaluate

$$24xy^{2}$$

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}.

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

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

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

Use the distributive property to multiply x^{2}-2xy by x+2y and combine like terms.

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

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

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

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

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

Combine 12xy^{2} and 4xy^{2} to get 16xy^{2}.

$$-6x^{2}y+16xy^{2}-8y^{3}+2xy\left(3x+4y\right)-\left(-2y\right)^{3}$$

Use the distributive property to multiply 2xy by 3x+4y.

$$-6x^{2}y+16xy^{2}-8y^{3}+6x^{2}y+8xy^{2}-\left(-2y\right)^{3}$$

Combine -6x^{2}y and 6x^{2}y to get 0.

$$16xy^{2}-8y^{3}+8xy^{2}-\left(-2y\right)^{3}$$

Combine 16xy^{2} and 8xy^{2} to get 24xy^{2}.

$$24xy^{2}-8y^{3}-\left(-2y\right)^{3}$$

Expand \left(-2y\right)^{3}.

$$24xy^{2}-8y^{3}-\left(-2\right)^{3}y^{3}$$

Calculate -2 to the power of 3 and get -8.

$$24xy^{2}-8y^{3}-\left(-8y^{3}\right)$$

The opposite of -8y^{3} is 8y^{3}.

$$24xy^{2}-8y^{3}+8y^{3}$$

Combine -8y^{3} and 8y^{3} to get 0.

$$24xy^{2}$$

Expand

$$24xy^{2}$$

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}.

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

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

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

Use the distributive property to multiply x^{2}-2xy by x+2y and combine like terms.

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

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

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

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

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

Combine 12xy^{2} and 4xy^{2} to get 16xy^{2}.

$$-6x^{2}y+16xy^{2}-8y^{3}+2xy\left(3x+4y\right)-\left(-2y\right)^{3}$$

Use the distributive property to multiply 2xy by 3x+4y.

$$-6x^{2}y+16xy^{2}-8y^{3}+6x^{2}y+8xy^{2}-\left(-2y\right)^{3}$$

Combine -6x^{2}y and 6x^{2}y to get 0.

$$16xy^{2}-8y^{3}+8xy^{2}-\left(-2y\right)^{3}$$

Combine 16xy^{2} and 8xy^{2} to get 24xy^{2}.

$$24xy^{2}-8y^{3}-\left(-2y\right)^{3}$$

Expand \left(-2y\right)^{3}.

$$24xy^{2}-8y^{3}-\left(-2\right)^{3}y^{3}$$

Calculate -2 to the power of 3 and get -8.

$$24xy^{2}-8y^{3}-\left(-8y^{3}\right)$$

The opposite of -8y^{3} is 8y^{3}.

$$24xy^{2}-8y^{3}+8y^{3}$$

Combine -8y^{3} and 8y^{3} to get 0.

$$24xy^{2}$$