(x+y)^4+x^4+y^4
=(x+y)^4+x^4+2x^2y^2+y^4-2x^2y^2
=(x+y)^4+(x^2+y^2)^2-2x^2y^2
=(x+y)^4+[(x^2+2xy+y^2)-2xy]^2-2x^2y^2
=(x+y)^4+[(x+y)^2-2xy]^2-2x^2y^2
=(x+y)^4+(x+y)^4-4xy(x+y)^2+4x^2y^2-2x^2y^2
=2(x+y)^4-4xy(x+y)^2+2x^2y^2
=2[(x+y)^4-2xy(x+y)^2+x^2y^2]
=2[(x+y)^2-xy]^2
=2(x^2+2xy+y^2-xy)^2
=2(x^2+xy+y^2)^2
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