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Find all functions f with f(x+y)+f(x)f(y)=f(xy)+(y+1)f(x)+(x+1)f(y)

Source: (4th Middle European Mathematical Olympiad, Individual Competition, Problem 1)

September 11, 2010
functionalgebrafunctional equationalgebra solved

Problem Statement

Find all functions f:R→Rf:\mathbb{R}\to\mathbb{R} such that for all x,y∈Rx, y\in\mathbb{R}, we have f(x+y)+f(x)f(y)=f(xy)+(y+1)f(x)+(x+1)f(y).f(x+y)+f(x)f(y)=f(xy)+(y+1)f(x)+(x+1)f(y).
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