10. (21 points) Find all functions $f: \mathbf{R} \rightarrow \mathbf{R}$, such that for any $x, y$ we have
$$
f(x+y)=f(x)+f(y)+2xy,
$$

and $x^{2}-|x|^{\frac{1}{2}} \leqslant f(x) \leqslant x^{2}+|x|^{\frac{1}{2}}$.
10. Let $g(x)=f(x)-x^{2}$. Then for any $x, y$ we have $g(x+y)=g(x)+g(y)$, and $|g(x)| \leqslant|x|^{\frac{1}{2}}$. Therefore, for any $x$ and positive integer $n$ we have $|g(x)|=\frac{|g(n x)|}{n} \leqslant \sqrt{\frac{|x|}{n}}$.
Let $n \rightarrow+\infty$, we get $g(x)=0$. Thus, $f(x)=x^{2}$.