$VT = x^4+y^4 +(x+y)^4 \\
= (x^2+y^2)^2 - 2x^2y^2 + (x+y)^4 \\
= [(x+y)^2-2xy]^2 - 2x^2y^2 + (x+y)^4 \\
= (x+y)^4 - 4xy(x+y)^2 + 4x^2y^2 - 2x^2y^2 + (x+y)^4 \\
= 2(x+y)^4 - 4xy(x+y)^2 + 2x^2y^2 \\
= 2[(x+y)^4 - 2xy(x+y)^2 + x^2y^2] \\
= 2[(x+y)^2-xy]^2 \\
= 2(x^2+xy+y^2)^2 = VP$