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 $x^{2}-|x|^{\frac{1}{2}} \leqslant f(x) \leqslant x^{2}+|x|^{\frac{1}{2}}$.