\(xy\left(x+y\right)+yz\left(y+z\right)+xz\left(x+z\right)+2xyz\)
\(=\left[xy\left(x+y\right)+xyz\right]+\left[yz\left(y+z\right)+xyz\right]+xz\left(x+z\right)\)
\(=xy\left(x+y+z\right)+yz\left(x+y+z\right)+xz\left(x+z\right)\)
\(=y\left(x+y+z\right)\left(x+z\right)+xz\left(x+z\right)\)
\(=\left(x+z\right)\left(x^2+y^2+yz+xz\right)\)
\(=\left(x+z\right)\left(x+y\right)\left(y+z\right)\)
\(xy\left(x+y\right)+yz\left(y+z\right)+xz\left(x+z\right)+2xyz.\)
\(=x^2y+xy^2+x^2z+xz^2+2xyz+yz\left(y+z\right)\)
\(=x^2\left(y+z\right)+x\left(y^2+z^2+2yz\right)+yz\left(y+z\right)\)
\(=x^2\left(y+z\right)+x\left(y+z\right)^2+yz\left(y+z\right)\)
\(=\left(y+z\right)\left(x^2+xy+xz+yz\right)\)
\(=\left(y+z\right)\left[x\left(x+z\right)+y\left(x+z\right)\right]=\left(y+z\right)\left(x+y\right)\left(x+z\right)\)
