$\begin{array}{l}x^4 + y^4 + (x+y)^4\\ = x^4 + y^4 + [(x+y)^2]^2\\ = x^4 + y^4 + (x^2 + 2xy + y^2)^2\\ = x^4 + y^4 + x^4 + 4x^2y^2 + y^4 + 4x^3y + 4xy^3 + 2x^2y^2\\ = 2(x^4 + 2x^3y + y^4 + 2xy^3 + x^2y^2 + 2x^2y^2)\\ = 2(x^2 + xy + y^2)^2\end{array}$