\(=\left(x+y\right)^3-3xy\left(x+y\right)+z^3-3xyz\\ =\left(x+y+z\right)\left(x^2+2xy+y^2-xz-yz+z^2\right)-3xy\left(x+y+z\right)\\ =\left(x+y+z\right)\left(x^2+y^2+z^2-xy-yz-zx\right)\)
\(x^3+y^3+z^3-3xyz\)
\(=\left(x^3+3x^2y+3xy^2+y^3\right)+z^3-3x^2y-3xy^2-3xyz\)
\(=\left(x+y\right)^3+z^3-3xy\left(x+y+z\right)\)
\(=\left(x+y+z\right)\left[\left(x+y\right)^2-\left(x+y\right)z+z^2\right]-3xy\left(x+y+z\right)\)
\(=\left(x+y+z\right)\left(x^2+2xy+y^2-xz-yz+z^2-3xy\right)\)
\(=\left(x+y+z\right)\left(x^2+y^2+z^2-xy-yz-xz\right)\)
x3 + y3 + z3 - 3xyz
= (x+y)3 - 3xy(x-y) + z3 - 3xyz
= [(x+y)3 + z3] - 3xy(x+y+z)
= (x+y+z)3 - 3z(x+y)(x+y+z) - 3xy(x-y-z)
= (x+y+z)[(x+y+z)2 - 3z(x+y) - 3xy]
= (x+y+z)(x2 + y2 + z2 + 2xy + 2xz + 2yz - 3xz - 3yz - 3xy)
= (x+y+z)(x2 + y2 + z2- xy - xz - yz)
