This is a generalization of
Solving the functional equation
When are there solutions to $f(x+y)-f(x-y) = axy+bx+cy+d $ (i.e., what restrictions are there on $a, b, c, d$) and what are they?
I have a solution, but want to see what others come up with.
An immediate observation taking $x=y=0$ is that $d$ must be $0$.
Next, with $y=0$, we find that $0=f(x)-f(x)=bx$ for all $x$, so that $b$ too must be $0$.
At this point, one may repeat DHMO's solution in the linked question to find that solutions do exist and are of the form
$$f(z)=\frac{a}4z^2+\frac{c}{2}z+f(0)$$
