3. Let the function be
$$
f(x)=\frac{\sqrt{2} \sin \left(x+\frac{\pi}{4}\right)+2 x^{2}+x}{2 x^{2}+\cos x}
$$

with the maximum value $M$ and the minimum value $m$. Then $M$ and $m$ satisfy ( ).
(A) $M+m=2$
(B) $M+m=4$
(C) $M-m=2$
(D) $M-m=4$
3.A.
$$
f(x)=\frac{\sin x+\cos x+2 x^{2}+x}{2 x^{2}+\cos x}=\frac{\sin x+x}{2 x^{2}+\cos x}+1 \text {. }
$$

Since $g(x)=\frac{\sin x+x}{2 x^{2}+\cos x}$ is an odd function, and $f(x)$ has a maximum value and a minimum value, therefore, $g(x)$ also has a maximum value $M'$ and a minimum value $m'$, and $M'+m'=0$.
Thus, $M+m=(M'+1)+(m'+1)=2$.