Question 251796
First, let's expand {{{(x^2+2xy-y)^2}}}

{{{(x^2+2xy-y)(x^2+2xy-y)}}} Rewrite {{{(x^2+2xy-y)^2}}} as {{{(x^2+2xy-y)(x^2+2xy-y)}}}

{{{x^2(x^2+2xy-y)+2xy(x^2+2xy-y)-y(x^2+2xy-y)}}} Expand. Recall that {{{(A+B+C)(D+E+F)=A(D+E+F)+B(D+E+F)+C(D+E+F)}}}

{{{(x^2)(x^2)+(x^2)(2xy)+(x^2)(-y)+(2xy)(x^2)+(2xy)(2xy)+(2xy)(-y)+(-y)(x^2)+(-y)(2xy)+(-y)(-y)}}} Distribute.

{{{x^4+2x^3y-x^2y+2x^3y+4x^2y^2-2xy^2-x^2y-2xy^2+y^2}}} Multiply.

{{{x^4+4x^3y-2x^2y+4x^2y^2-4xy^2+y^2}}} Now combine like terms.

So {{{(x^2+2xy-y)(x^2+2xy-y)}}} expands to {{{x^4+4x^3y-2x^2y+4x^2y^2-4xy^2+y^2}}}.

In other words, {{{(x^2+2xy-y)(x^2+2xy-y)=x^4+4x^3y-2x^2y+4x^2y^2-4xy^2+y^2}}}.

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This means that {{{(x+y)(x^2+2xy-y)^2}}} then becomes {{{(x+y)(x^4+4x^3y-2x^2y+4x^2y^2-4xy^2+y^2)}}}

Now let's finish up...

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

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

{{{x^5+4x^4y-2x^3y+4x^3y^2-4x^2y^2+xy^2+x^4y+4x^3y^2-2x^2y^2+4y^3x^2-4y^3x+y^3}}} Multiply.

{{{x^5+5x^4y-2x^3y+8x^3y^2-6x^2y^2+xy^2+4y^3x^2-4y^3x+y^3}}} Now combine like terms.

So {{{(x+y)(x^4+4x^3y-2x^2y+4x^2y^2-4xy^2+y^2)}}} expands to {{{x^5+5x^4y-2x^3y+8x^3y^2-6x^2y^2+xy^2+4y^3x^2-4y^3x+y^3}}}.

In other words, {{{(x+y)(x^4+4x^3y-2x^2y+4x^2y^2-4xy^2+y^2)=x^5+5x^4y-2x^3y+8x^3y^2-6x^2y^2+xy^2+4y^3x^2-4y^3x+y^3}}}.

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