KöMaL 2006 A390

Find all functions $f\colon \mathbb{R}\to\mathbb{R}$ such that ( $f$($x$)+$f$($y$) ) ( $f$($z$)+1 ) =$f$($xz$-$y$)+$f$($x$+$yz$) for all real $x$, $y$, $z$.