<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] $$x^4 + y^4 + (x + y)^4
=2 x^4 + 4 x^3 y + 6 x^2 y^2 + 4 x y^3 + 2 y^4$$
$$=x (x (x (2 x + 4 y) + 6 y^2) + 4 y^3) + 2 y^4
=2 x^4 + y (4 x^3 + y (6 x^2 + y (4 x + 2 y)))$$
$$=4 y^2 (x^2 + x y) + 2 (x^2 + x y)^2 + 2 y^4
=2 (x^4 + 2 x^3 y + 3 x^2 y^2 + 2 x y^3 + y^4)$$
$$=2 (x^2 + x y + y^2)^2$$
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