Example 7 Find the product of all roots of the equation $z^{3}+|z|^{2}=10 \mathrm{i}$.

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Solve:
$$
\begin{aligned}
& z^{3}=-|z|^{2}+10 \mathrm{i} . \\
\Rightarrow & \bar{z}^{3}=-|z|^{2}-10 \mathrm{i} \\
\Rightarrow & |z|^{6}=|z|^{4}+100 \\
\Rightarrow & |z|^{2}=5 .
\end{aligned}
$$

Thus, the original equation becomes $z^{3}+5-10 \mathrm{i}=0$, and the product of all roots is $-5+10 \mathrm{i}$.