Solve (y-2x)^3-(2x-y)(-2x-y)-(2x+y)^3-[-4x(4x^2+3y^2)]

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

Evaluate

$$4x^{2}-y^{2}$$

Solution Steps

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

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

Use the distributive property to multiply 2x-y by -2x-y and combine like terms.

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

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

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

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

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

To find the opposite of 8x^{3}+12x^{2}y+6xy^{2}+y^{3}, find the opposite of each term.

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

Combine -8x^{3} and -8x^{3} to get -16x^{3}.

$$y^{3}-6y^{2}x+12yx^{2}-16x^{3}+4x^{2}-y^{2}-12x^{2}y-6xy^{2}-y^{3}-\left(-4x\left(4x^{2}+3y^{2}\right)\right)$$

Combine 12yx^{2} and -12x^{2}y to get 0.

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

Combine -6y^{2}x and -6xy^{2} to get -12y^{2}x.

$$y^{3}-12y^{2}x-16x^{3}+4x^{2}-y^{2}-y^{3}-\left(-4x\left(4x^{2}+3y^{2}\right)\right)$$

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

$$-12y^{2}x-16x^{3}+4x^{2}-y^{2}-\left(-4x\left(4x^{2}+3y^{2}\right)\right)$$

The opposite of -4x\left(4x^{2}+3y^{2}\right) is 4x\left(4x^{2}+3y^{2}\right).

$$-12y^{2}x-16x^{3}+4x^{2}-y^{2}+4x\left(4x^{2}+3y^{2}\right)$$

Use the distributive property to multiply 4x by 4x^{2}+3y^{2}.

$$-12y^{2}x-16x^{3}+4x^{2}-y^{2}+16x^{3}+12xy^{2}$$

Combine -16x^{3} and 16x^{3} to get 0.

$$-12y^{2}x+4x^{2}-y^{2}+12xy^{2}$$

Combine -12y^{2}x and 12xy^{2} to get 0.

$$4x^{2}-y^{2}$$

Expand

$$4x^{2}-y^{2}$$

Solution Steps

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

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

Use the distributive property to multiply 2x-y by -2x-y and combine like terms.

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

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

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

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

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

To find the opposite of 8x^{3}+12x^{2}y+6xy^{2}+y^{3}, find the opposite of each term.

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

Combine -8x^{3} and -8x^{3} to get -16x^{3}.

$$y^{3}-6y^{2}x+12yx^{2}-16x^{3}+4x^{2}-y^{2}-12x^{2}y-6xy^{2}-y^{3}-\left(-4x\left(4x^{2}+3y^{2}\right)\right)$$

Combine 12yx^{2} and -12x^{2}y to get 0.

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

Combine -6y^{2}x and -6xy^{2} to get -12y^{2}x.

$$y^{3}-12y^{2}x-16x^{3}+4x^{2}-y^{2}-y^{3}-\left(-4x\left(4x^{2}+3y^{2}\right)\right)$$

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

$$-12y^{2}x-16x^{3}+4x^{2}-y^{2}-\left(-4x\left(4x^{2}+3y^{2}\right)\right)$$

The opposite of -4x\left(4x^{2}+3y^{2}\right) is 4x\left(4x^{2}+3y^{2}\right).

$$-12y^{2}x-16x^{3}+4x^{2}-y^{2}+4x\left(4x^{2}+3y^{2}\right)$$

Use the distributive property to multiply 4x by 4x^{2}+3y^{2}.

$$-12y^{2}x-16x^{3}+4x^{2}-y^{2}+16x^{3}+12xy^{2}$$

Combine -16x^{3} and 16x^{3} to get 0.

$$-12y^{2}x+4x^{2}-y^{2}+12xy^{2}$$

Combine -12y^{2}x and 12xy^{2} to get 0.

$$4x^{2}-y^{2}$$