Finding $|f(4)|$ given that $f$ is a continuous function satisfying $f(x)+f(2x+y)+5xy=f(3x-y)+2x^2+1\forall x,y\in\mathbb{R}$

Question

The question is simply to find $|f(4)|$ given that $f$ is a continuous function and satisfies the following functional equation $\forall x,y \in \mathbb{R}$.

$$f(x)+f(2x+y)+5xy=f(3x-y)+2x^2+1\forall x,y\in\mathbb{R}$$

Here is what I have done so far. If we put $x=y=0$, we get that $f(0)=1$ and if we let $x=0$, we can find that $f(x)$ is an even function. Any ideas on how to proceed. Thanks.

Answer 1

Hint Take an arbitrary $x$ and pick $y$ such that $2x+y=3x-y$.