Determine all pairs of functions from the set of real numbers to itsels that satisfy for all real numbers and
Proposed by Japan
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Determine all pairs $(f,g)$ of functions from the set of real numbers to itsels that satisfy $$g(f(x+y)) = f(x) + (2x + y)g(y)$$ for all real numbers $x$ and $y.$
Proposed by Japan
Source: Međunarodna matematička olimpijada, shortlist 2011
Školjka 
of functions from the set of real numbers to itsels that satisfy 
for all real numbers 
and 