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