9. (14 points) Given $f(x)=\sin 2 x+3 \sin x+$ $3 \cos x(0 \leqslant x<2 \pi)$. Find:
(1) the range of $f(x)$;
(2) the intervals of monotonicity of $f(x)$.
II. 9. (1) Let
$$
\sin x + \cos x = t \quad (-\sqrt{2} \leqslant t \leqslant \sqrt{2}).
$$

Then $\sin 2x = t^2 - 1$.
We need to find the range of $g(t) = t^2 + 3t - 1$.
Since $g(t) = \left(t + \frac{3}{2}\right)^2 - \frac{13}{4}$, when $t = \pm \sqrt{2}$,
$g(t)$ attains its extreme values, i.e., the range of $f(x)$ is
$$
[1 - 3\sqrt{2}, 1 + 3\sqrt{2}].
$$
(2) $f'(x) = 2 \cos 2x + 3(\cos x - \sin x)$
$$
= (\cos x - \sin x)(2 \cos x + 2 \sin x + 3).
$$

Since $2 \cos x + 2 \sin x + 3 \geqslant 0$, the decreasing interval of $f(x)$ is $\left[\frac{\pi}{4}, \frac{5\pi}{4}\right]$, and the increasing intervals are $\left[0, \frac{\pi}{4}\right]$ and $\left[\frac{5\pi}{4}, 2\pi\right]$.