(x+y+z)^3-x^3-y^3-z^3
=(x+y+z)^3-x^3-(y^3+z^3)
=(x+y+z-x)[(x+y+z)^2+x(x+y+z)+x^2]-(y+z)(y^2-yz+z^2)
=(y+z)(x^2+y^2+z^2+2xy+2yz+2xz+x^2+xy+xz+x^2-y^2+yz-z^2)
=(y+z)(3x^2+3xy+3yz+3xz)
=3(y+z)[x(x+y)+z(x+y)]
=3(y+z)(x+z)(x+y)