According to the question mentioned here, it seems that there is no function $f(x)$ such that the functional equation
can hold. Motivated by this question, I found it interesting to somehow extend the question.
What conditions are required for a given function $g(x)$ such that there exists a function $f(x)$ that can satisfy the following equality $$f(x+y)=f(x)+f(y)+g(xy)$$ where $f(x)$ and $g(x)$ are real valued functions of real variable.
Any hint or help is appreciated. :)