For real numbers $a$ and $b$, we define $a \nabla b=a b-b a^{2}$. For example, $5 \nabla 4=5(4)-4\left(5^{2}\right)=-80$. Determine the sum of the values of $x$ for which $(2 \nabla x)-8=x \nabla 6$.
Using the given definition, the following equations are equivalent:

$$
\begin{aligned}
(2 \nabla x)-8 & =x \nabla 6 \\
(2 x-4 x)-8 & =6 x-6 x^{2} \\
6 x^{2}-8 x-8 & =0 \\
3 x^{2}-4 x-4 & =0
\end{aligned}
$$

The sum of the values of $x$ that satisfy the original equation equals the sum of the roots of this quadratic equation.

This sum equals $-\frac{-4}{3}$ or $\frac{4}{3}$.

(We could calculate the roots and add these, or use the fact that the sum of the roots of the quadratic equation $a x^{2}+b x+c=0$ is $-\frac{b}{a}$.)