On generalizations of the Pompeiu functional equation (Q1380965)

In addition to the functional equation \(f(ax+by+cxy) = f(x)+f(y)+f(x)f(y)\) (the variables go through all real numbers; \(f\) is real valued), where the solutions are immediate after distinguishing between the cases \(f(0)=0\) and \(f(0)=-1\) and between \(c=0\) and \(c\neq 0\), the authors offer also the general solution of \(f(x+y+xy)=p(x)+q(y)+g(x)h(y)\) (the variables take all real values except \(-1\); \(f,p,q,g,h\) are real valued unknown functions).