Does a function $f$ that satisfies $f(2x+y)=f(x+1)+f(y)+4xy$ for all $x,y$ exist?

Question

Does a function $f$ that satisfies $f(2x+y)=f(x+1)+f(y)+4xy$ for all $x,y$ exist?

I was able to figure out some specific values of such a function by plugging in certain numbers, for instance using $x=0,y=1$ I found that $f(1)=0$. Similarly, I found the values of $f(0)$ by using $x=1,y=0$ and $f(2)$ using $x=-1,y=2$. But I can't find the entire function...

Answer 1

Suppose such $f$ exists. I found that $f(0) = 0, f(1) = 0, f(2) = 8$.

With $x=1, y=1$, $$ f(3) = f(2) + f(1) + 4 = 12$$

With $x=1, y=2$, $$f(4) = f(2) + f(2) + 8 = 24 $$

With $x=2, y=0$, $$ f(4) = f(3) + f(0) + 0 = f(3) = 12$$

Contradiction.