Let $f(x)$ be a polynomial such that $f(f(x))=f(x-1)f'(x+1)$ and $f'(0)=2$. Find $f(x)$. 
Let $f(x)$ be a polynomial such that $f(f(x))=f(x-1)f'(x+1)$ and $f'(0)=2$. Find $f(x)$.

Putting $x=-1$, I get $f(f(-1))=f(-2)f'(0)=2f(-2)$
Putting $x=0$, I get $f(f(0))=f(-1)f'(1)$
Also, replacing $x$ by $x+1$, I get $f(f(x+1))=f(x)f'(x+2)$
Not sure if I am getting anywhere.

Hint:
$\deg f(x) = n$ then $\deg f(f(x)) = n^2$ and $\deg f(x-1)f'(x+1) = n + (n-1)$. Hence, $n^2 = n+(n-1)$
