$x^2+y^2+2\mathrm{xy}-z^2-x-y+z=$

$=(x^2+y^2+2\mathrm{xy})-z^2-\left(x+y\right)+z=$

$={\left(x+y\right)}^2-z^2-\left(\left(x+y\right)-z\right)=$

$=\left(x+y+z\right)·\left(x+y-z\right)-\left(x+y-z\right)=$

$=\left(x+y-z\right)·\left(x+y+z-1\right)$