Find all functions $f: mathbb{R}rightarrow mathbb{R}$ satisfying a functional equation

Find all functions $f: \mathbb{R}\rightarrow \mathbb{R}$ satisfying:
$f\left ( x \right )f\left ( y \right )+ f\left ( xy \right )+ f\left ( x \right )+f\left ( y \right )= f\left ( x+y \right )+ 2\,xy$

I tried the standard way: $x=0, x=y, x=1,...$ but without any success. I spent quite some time trying to solve it but didn't succeed.

I tried to reduce it to Cauchy's 1-4 equations but didn't succeed. In the corse of it, I found interesting works of Aczel, Erdos and even Putnum, but they are not directly related, I guess.

Any idea? I am interested in this problem but I couldn't solve!