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