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A functional equation

Source: Switzerland Math Olympiad, Final round 2014, P3

June 2, 2014
functioninductionalgebra proposedalgebra

Problem Statement

Find all such functions f:R→Rf :\mathbb{R}\to \mathbb{R} such that for all x,y∈Rx,y\in\mathbb{R} the following holds : f(x2)+f(xy)=f(x)f(y)+yf(x)+xf(x+y) f(x^2)+f(xy)=f(x)f(y)+yf(x)+xf(x+y)
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