<s>[INST] Factor $(x+y)^4+x^4+y^4$ Title says it all, I just want to know how to factor $(x+y)^4+x^4+y^4$. I only know that it's possible to factor, but got no idea how to do it. If it were a single-variable polynomial I could try to find rational roots or something, but I'm lost with this one.
 [/INST] $$(x+y)^4+x^4+y^4=(x^2+y^2+2xy)^2+(x^2+y^2)^2-2(xy)^2=\\
[(x^2+y^2+2xy)^2-(xy)^2]+[(x^2+y^2)^2-(xy)^2]=\\
(x^2+y^2+xy)(x^2+y^2+3xy)+(x^2+y^2+xy)(x^2+y^2-xy)=\\
(x^2+y^2+xy)(2x^2+2y^2+2xy)=2(x^2+y^2+xy)^2$$
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