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Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $$f(f(x)^2+f(y))=xf(x)+y$$ for all $x,y\in{\mathbb{R}}$.

Here is my approach to the problem:

We see that $f(x)=x$ is an obvious solution (Just trying easy linear equations). I think this would be the only solution to the problem.

Am I right? And how to prove that there is no other solution? (Note: I am a beginner at functional equations)