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

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

Evaluate

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Solution Steps

Square x^{3}-y^{3}+3xy.

$$x^{6}-2x^{3}y^{3}+9x^{2}y^{2}-6xy^{4}+y^{6}+6yx^{4}-\left(x^{3}+y^{3}+3xy\right)^{2}+4xy^{3}\left(x^{2}+3y\right)$$

Square x^{3}+y^{3}+3xy.

$$x^{6}-2x^{3}y^{3}+9x^{2}y^{2}-6xy^{4}+y^{6}+6yx^{4}-\left(x^{6}+2x^{3}y^{3}+9x^{2}y^{2}+6xy^{4}+y^{6}+6yx^{4}\right)+4xy^{3}\left(x^{2}+3y\right)$$

To find the opposite of x^{6}+2x^{3}y^{3}+9x^{2}y^{2}+6xy^{4}+y^{6}+6yx^{4}, find the opposite of each term.

$$x^{6}-2x^{3}y^{3}+9x^{2}y^{2}-6xy^{4}+y^{6}+6yx^{4}-x^{6}-2x^{3}y^{3}-9x^{2}y^{2}-6xy^{4}-y^{6}-6yx^{4}+4xy^{3}\left(x^{2}+3y\right)$$

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

$$-2x^{3}y^{3}+9x^{2}y^{2}-6xy^{4}+y^{6}+6yx^{4}-2x^{3}y^{3}-9x^{2}y^{2}-6xy^{4}-y^{6}-6yx^{4}+4xy^{3}\left(x^{2}+3y\right)$$

Combine -2x^{3}y^{3} and -2x^{3}y^{3} to get -4x^{3}y^{3}.

$$-4x^{3}y^{3}+9x^{2}y^{2}-6xy^{4}+y^{6}+6yx^{4}-9x^{2}y^{2}-6xy^{4}-y^{6}-6yx^{4}+4xy^{3}\left(x^{2}+3y\right)$$

Combine 9x^{2}y^{2} and -9x^{2}y^{2} to get 0.

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

Combine -6xy^{4} and -6xy^{4} to get -12xy^{4}.

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

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

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

Combine 6yx^{4} and -6yx^{4} to get 0.

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

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

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

Combine -4x^{3}y^{3} and 4y^{3}x^{3} to get 0.

$$-12xy^{4}+12xy^{4}$$

Combine -12xy^{4} and 12xy^{4} to get 0.

$$0$$

Factor

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