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Weird looking function

Source: European Mathematical Cup 2014, Senior Division, P4

December 14, 2014
functionalgebra

Problem Statement

Find all functions f:R→R f:\mathbb{R}\rightarrow\mathbb{R} such that for all x,y∈Rx,y\in{{\mathbb{R}}} holds f(x2)+f(2y2)=(f(x+y)+f(y))(f(x−y)+f(y))f(x^2)+f(2y^2)=(f(x+y)+f(y))(f(x-y)+f(y))
Proposed by Matija Bucić
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