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


{{{(x^2+2xy-y)^2}}} Start with the given expression.



{{{(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+y)(x^4+4x^3y-2x^2y+4x^2y^2-4xy^2+y^2)}}} Start with the given expression.



{{{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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Answer:



So {{{(x+y)(x^2+2xy-y)^2}}} fully expands out 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^2+2xy-y)^2=x^5+5x^4y-2x^3y+8x^3y^2-6x^2y^2+xy^2+4y^3x^2-4y^3x+y^3}}}




Note: if this problem was more complicated than it should have been, then maybe there's a typo somewhere.