INMO 2011 Problem 6

Find all functions $f: \mathbf{R} \rightarrow \mathbf{R}$ such that

$$f(x+y) f(x-y)=(f(x)+f(y))^{2}-4 x^{2} f(y)$$

for all $x, y \in \mathbf{R}$, where $\mathbf{R}$ denotes the set of all real numbers.