Solve [(-x+y)^{2}-(x+y)^2][(-x+y)^2+(-x+y)(x+y)+(x+y)^2]+12xy^3

$$[ ( - x + y ) ^ { 2 } - ( x + y ) ^ { 2 } ] [ ( - x + y ) ^ { 2 } + ( - x + y ) ( x + y ) + ( x + y ) ^ { 2 } ] + 12 x y ^ { 3 }$$

Evaluate

$$-4yx^{3}$$

Solution Steps

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

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

Calculate -x to the power of 2 and get x^{2}.

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

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

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

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

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

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

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

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

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

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

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

Calculate -x to the power of 2 and get x^{2}.

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

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

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

Combine 2\left(-x\right)y and \left(-x\right)y to get 3\left(-x\right)y.

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

Combine y^{2} and y^{2} to get 2y^{2}.

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

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

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

Combine x^{2} and x^{2} to get 2x^{2}.

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

Combine yx and 2xy to get 3yx.

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

Combine 2y^{2} and y^{2} to get 3y^{2}.

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

Use the distributive property to multiply 2\left(-x\right)y-2xy by 2x^{2}+3\left(-x\right)y+3y^{2}+\left(-x\right)x+3yx and combine like terms.

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

Calculate -x to the power of 2 and get x^{2}.

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

Calculate -x to the power of 2 and get x^{2}.

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

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

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

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

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

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

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

Combine -6xy^{3} and 12xy^{3} to get 6xy^{3}.

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

Multiply 2 and -1 to get -2.

$$-2xyx^{2}+6\left(-1\right)xy^{3}-2yx^{3}+6xy^{3}$$

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

$$-2x^{3}y+6\left(-1\right)xy^{3}-2yx^{3}+6xy^{3}$$

Multiply 6 and -1 to get -6.

$$-2x^{3}y-6xy^{3}-2yx^{3}+6xy^{3}$$

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

$$-4x^{3}y-6xy^{3}+6xy^{3}$$

Combine -6xy^{3} and 6xy^{3} to get 0.

$$-4x^{3}y$$

Expand

$$-4yx^{3}$$

Solution Steps

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

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

Calculate -x to the power of 2 and get x^{2}.

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

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

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

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

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

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

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

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

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

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

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

Calculate -x to the power of 2 and get x^{2}.

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

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

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

Combine 2\left(-x\right)y and \left(-x\right)y to get 3\left(-x\right)y.

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

Combine y^{2} and y^{2} to get 2y^{2}.

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

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

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

Combine x^{2} and x^{2} to get 2x^{2}.

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

Combine yx and 2xy to get 3yx.

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

Combine 2y^{2} and y^{2} to get 3y^{2}.

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

Use the distributive property to multiply 2\left(-x\right)y-2xy by 2x^{2}+3\left(-x\right)y+3y^{2}+\left(-x\right)x+3yx and combine like terms.

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

Calculate -x to the power of 2 and get x^{2}.

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

Calculate -x to the power of 2 and get x^{2}.

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

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

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

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

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

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

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

Combine -6xy^{3} and 12xy^{3} to get 6xy^{3}.

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

Multiply 2 and -1 to get -2.

$$-2xyx^{2}+6\left(-1\right)xy^{3}-2yx^{3}+6xy^{3}$$

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

$$-2x^{3}y+6\left(-1\right)xy^{3}-2yx^{3}+6xy^{3}$$

Multiply 6 and -1 to get -6.

$$-2x^{3}y-6xy^{3}-2yx^{3}+6xy^{3}$$

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

$$-4x^{3}y-6xy^{3}+6xy^{3}$$

Combine -6xy^{3} and 6xy^{3} to get 0.

$$-4x^{3}y$$