Solution of the functional equation $f(x)f(y)=f(x+y)+xy$

Question

I have been working on problem sets training for the mathematical olympiad, and I was examining the above functional equation: $f(x)f(y)=f(x+y)+xy$. I am well aware it has an exceedingly elegant answer under: Find all functions $ f : \Bbb R \to \Bbb R$ which satisfy $f(x)f(y) = f(x + y) + xy$ for all $x, y ∈ \Bbb R$., but this is not my question. Manipulating, one can get: $f(x)f(1)=f(x+1)-x$. Let us set $f(1)=c$, then we have a linear difference, thus f can be at most quadratic: $cf(x)-f(x+1)=x$. If we solve for the coefficients of this quadratic, we can obtain the two solutions. While I was doing this, though, I was beginning to get certain doubts about this method. It seems to be not generally applicable without certain conditions, such as differentiability, allowing the taylor series expansion and legitimizing the approach. Is it valid in general? Thank you