\(xy\left(x+y\right)+yz\left(y+z\right)+xz\left(x+z\right)+2xyz\)

= \(xy\left(x+y\right)+yz\left(y+z\right)+xyz+xz\left(x+z\right)+xyz\)

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

= \(xy\left(x+y\right)+yz\left(x+y+z\right)+xz\left(x+y+z\right)\)

= \(xy\left(x+y\right)+\left(x+y+z\right)\left(yz+xz\right)\)

= \(xy\left(x+y\right)+\left(x+y+z\right).z\left(x+y\right)\)

= \(xy\left(x+y\right)+\left(xz+yz+z^2\right)\left(x+y\right)\)

= \(\left(x+y\right)\left(xy+yz+xz+z^2\right)\)

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

= \(\left(x+y\right)[y\left(x+z)+z(x+z\right)]\)

= \(\left(x+y\right)\left(x+z\right)\left(y+z\right)\)