10. Let the function $f(x)=\frac{(x+1)^{2}+\ln \left(\sqrt{x^{2}+1}+x\right)}{x^{2}+1}$ have a maximum value of $M$ and a minimum value of $N$. Determine the value of $M+N$.
$$
f(x)=\frac{x^{2}+1+2 x+\ln \left(\sqrt{x^{2}+1}+x\right)}{x^{2}+1}=1+\frac{2 x+\ln \left(\sqrt{x^{2}+1}+x\right)}{x^{2}+1} \text {, }
$$

Notice that $g(x)=\frac{2 x+\ln \left(\sqrt{x^{2}+1}+x\right)}{x^{2}+1}$ is an odd function, so $g(x)_{\text {min }}=-g(x)_{\max }$,
thus $M=1+g(x)_{\max }, N=1-g(x)_{\max } \Rightarrow M+N=2$.