Example 13 Find the sum of all negative roots of the equation $x^{3}-\frac{3}{2} \sqrt[3]{6} x^{2}+3=0$.
(2nd Hope Cup Mathematics Competition)
Solution: The original equation can be transformed into
$$
2 x^{3}-3 \sqrt[3]{6} x^{2}+6=0 \text {. }
$$

Let $\sqrt[3]{6}=y$, then the original equation can be transformed into
$$
2 x^{3}-3 x^{2} y+y^{3}=0 \text {, }
$$

which is $(x-y)^{2}(2 x+y)=0$.
Thus, $x=y$ or $x=-\frac{1}{2} y$,
i.e., $x=\sqrt[3]{6}$ or $x=-\frac{1}{2} \sqrt[3]{6}$.
$\therefore$ The sum of the negative roots of the original equation is $-\frac{1}{2} \sqrt[3]{6}$.
(4) Using the average value substitution. When the sum of several algebraic expressions in the equation differs by a constant, set their average value as a new unknown, which can simplify the equation.