Question

(x+y+z)³-(y+z-x)³-(z+x-y)³-(x+y-z)³

Answer 1

Observe that,

(x+y+z)3−(x+y−z)3=2z((x+y+z)2+(x+y−z)2)+(x+y)2−z2)=2z(3(x+y)2+z2)(1)

and,

(x−y−z)3−(x−y+z)3=−2z((x−y−z)2+(x−y+z)2)+(x−y)2−z2)=2z(3(x−y)2+z2)(2)

Therefore,

(x+y+z)3+(x−y−z)3−(x−y+z)3−(x+y−z)3=((x+y+z)3−(x+y−z)3)+((x−y−z)3−(x−y+z)3)=2z(3(x+y)2+z2)−2z(3(x−y)2+z2)=2z(3⋅4xy)=24xy