\(xyz-\left(xy+yz+xz\right)+\left(x+y+z\right)-1\)
\(=\left(xyz-xy-xz+x\right)-yz+y+z-1\)
\(=x\left(yz-y-z+1\right)-\left(yz-y-z+1\right)\)
\(=\left(x-1\right)\left(yz-y-z+1\right)\)
\(=\left(x-1\right)\left[y\left(z-1\right)-\left(z-1\right)\right]\)
\(=\left(x-1\right)\left(y-1\right)\left(z-1\right)\)