5.1.3 ** Given a complex number $z$ satisfying $|z|-1$, find the maximum value of $\left|z^{3}-3 z-2\right|$.

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Given that $\left|z^{3}-3 z-2\right|=\left|(z+1)^{2}(z-2)\right|=|z+1|^{2} \cdot|z-2|$, let $z=a+b \mathrm{i}(a, b \in \mathbf{R})$, then $a^{2}+b^{2}=1$.
$$
\begin{aligned}
\text { The original expression } & =\left|z^{3}-3 z+2\right|=|z+1|^{2} \cdot|z-2|=|(a+1)+b i|^{2} \cdot|(a-2)+b i| \\
& =2(a+1) \cdot \sqrt{5-4 a}=\sqrt{(2 a+2)(2 a+2)(5-4 a)} \\
& \leqslant \sqrt{\left(\frac{2 a+2+2 a+2+5-4 a}{3}\right)^{3}}=3 \sqrt{3},
\end{aligned}
$$

Equality holds if and only if $2 a+2=5-4 a$, i.e., $a=\frac{1}{2}, b= \pm \frac{\sqrt{3}}{2}$. Therefore, the maximum value sought is $3 \sqrt{3}$.