Example 7. Find the sum of all negative roots of the equation $x^{3}-\frac{3}{2} \sqrt[3]{6} x^{2}+3=0$. (Second Hope Cup Mathematics Competition)
To solve the equation with a zero root, the equation can be transformed as follows:
$$
\begin{aligned}
x+\frac{3}{x^{2}} & =\frac{3}{2} \sqrt[3]{6} . \\
x+\frac{3}{x^{2}} & =\frac{x}{2}+\frac{x}{2}+\frac{3}{x^{2}} \\
& \geqslant \sqrt[3]{\frac{3}{4}}=\frac{3}{2} \sqrt[3]{6} .
\end{aligned}
$$

From (1), we know that equality holds only when $\frac{x}{2}=\frac{3}{x^{2}}$. Therefore, $x=\sqrt[3]{6}$ is a positive root of the original equation. Thus, the original equation can be transformed into
$$
(x-\sqrt[3]{6})(x-\sqrt[3]{6})\left(x+\frac{\sqrt[3]{6}}{2}\right):=0 .
$$

Therefore, the only negative root is $x=-\frac{\sqrt[3]{6}}{2}$.