\(\left(x+y+z\right)^3-x^3-y^3-z^3\)
\(=x^3+y^3+z^3+3x^2y+3xy^2+3x^2z+3xz^2+3y^2z+3yz^2+6xyz\)\(=3\left(xyz+x^2y+x^2z+xz^2+xy^2+y^2z+xyz+yz^2\right)\)
\(=3\left[xy\left(x+z\right)+xz\left(x+z\right)+y^2\left(x+z\right)+yz\left(x+z\right)\right]\)
\(=3\left(x+z\right)\left[xy+xz+y^2+yz\right]\)
\(=3\left(x+z\right)\left[x\left(y+z\right)+y\left(y+z\right)\right]\)
\(=3\left(x+z\right)\left(y+z\right)\left(x+y\right)\)
\(\left(x+y+z\right)^3-x^3-y^3-z^3\\ \\=\left[x^3+y^3+z^3+3\left(x+y\right)\left(x+z\right)\left(y+z\right)\right]-\left(x^3+y^3+z^3\right)\\ \\=3\left(x+y\right)\left(x+z\right)\left(y+z\right)\)
(x+y+z)3−x3−y3−z3=[x3+y3+z3+3(x+y)(x+z)(y+z)]−(x3+y3+z3)=3(x+y)(x+z)(y+z)
