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