Question

(x+y)^3+(y+z)^3+(z+x)^3-3(x+y)(y+z)(z+x)=2(x^3+y^3+z^3-3xyz)

Answer 1

In this question formula [a3+b3+c3 - 3abc = (a+b+c)(a2+b2+c2 - ab-bc-ca)] is used.

LHS part

(x+y)3 + (y+z)3 + (z+x)3 - 3(x+y)(y+z)(z+x)

=(x+y + y+z + z+x){(x+y)2 + (y+z)2 + (z+x)2 - (x+y)(y+z) - (y+z)(z+x) - (z+x)(x+y)}

=2(x+y+z){x2+y2+2xy + y2+z2+2yz + z2+x2+2xz - xy-y2-xz-yz - yz-xy-z2-xz - xz-x2-yz-xy}

=2(x+y+z)(x2 + y2 + z2 - xy - yz- zx)

2(x3+ y3+ z3 - 3xyz)

=RHS part

Proved