Functional equation with only two solutions?

Recently I started studying functional equations. Now I'm trying to find all solutions to the following functional equation:
$$(f(2x))^3=f(4x)((f(x))^2+xf(x)).$$
Unfortunately, I was able to show only few things about this equation, moreover, I wasn't even able to find $f(0)$ using substitution method. What I got (only by guessing) that there are at least two solutions: $f(x)=0$ and $f(x)=x$. But I don't know if there are any others. Any help would be very appreciated.