<s>[INST] Algebra (not so simple) Factoring I got stuck on this problem from my Math Challenge II Algebra Class:
Factorize the following: $$(x^2+xy+y^2)^2-4xy(x^2+y^2)$$
Hint: Let $u=x+y$ and $v=xy$.
Here's what I did: $$(x^2+xy+y^2)^2-4xy(x^2+y^2)$$ Convert into terms of $u$ and $v$: $$(u^2-v)^2-4v(u^2-2v)$$ $$(u^2-v)^2-4u^2v+8v^2$$ $$u^4-2u^2v+v^2-4u^2v-8v$$ $$u^4-6u^2v-7v^2$$ $$(u^2-7v)(u^2+v)$$ Then convert back into terms of $x$ and $y$: $$(x^2-5xy+y^2)(x^2+3xy+y^2)$$ When I expand the original equation, I get: $$x^4-2x^3y+3x^2y^2-2xy^3+y^4$$ When I expand the simplified result, I get: $$x^4-2x^3y-13x^2y^2-2xy^3+y^4$$ What did I do wrong?
EDIT: Thanks for explaining it to me. I won't edit the actual question, but I'll put corrections here. $$(u^2-v)^2-4v(u^2-2v)$$ $$(u^2-v)^2-4u^2v+8v^2$$ $$u^4-2u^2v+v^2-4u^2v+8v$$ $$u^4-6u^2v+9v^2$$ $$(u^2-3v)^2$$ $$(x^2-xy+y^2)^2$$ Please correct me if I made another mistake (I'm prone to mistakes).
 [/INST] See you can do even like this: $$(x^2+xy+y^2)^2-4xy(x^2+y^2)=(x^2+y^2)^2+x^2y^2+2xy(x^2+y^2)-4xy(x^2+y^2)=(x^2+y^2)^2+x^2y^2-2xy(x^2+y^2)=(x^2+y^2-xy)^2$$
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