\(x\left(y^3-z^3\right)+y\left(z^3-x^3\right)+z\left(x^3-y^3\right)\)
\(=y^3\left(x-z\right)+z^3\left(y-x\right)+x^3\left(z-y\right)\)
\(=y^3\left(x-y-z+y\right)+z^3\left(y-x\right)+x^3\left(z-y\right)\)
\(=\left(x-y\right)\left(y^3-z^3\right)+\left(y-z\right)\left(y^3-x^3\right)\)
\(=\left(x-y\right)\left(y-z\right)\left(y^2+yz+z^2\right)-\left(y-z\right)\left(x-y\right)\left(x^2+xy+y^2\right)\)
\(=\left(x-y\right)\left(y-z\right)\left(y^2+yz+z^2-x^2-xy-y^2\right)\)
\(=\left(x-y\right)\left(y-z\right)\left[y\left(z-x\right)+\left(z-x\right)\left(z+x\right)\right]\)
\(=\left(x-y\right)\left(y-z\right)\left(z-x\right)\left(x+y+z\right)\)
