8. Find all values of the parameter $a$ for which the quadratic trinomial $\frac{1}{3} x^{2}+\left(a+\frac{1}{2}\right) x+\left(a^{2}+a\right)$ has two roots, the sum of the cubes of which is exactly three times their product. In your answer, specify the largest such $a$.
Answer: $a=-1 / 4$.

Solution: By Vieta's theorem, we express the equality $x_{1}^{3}+x_{2}^{3}=3 x_{1} x_{2}$ in terms of $a$, from which a quadratic equation in $a$ is obtained, and two options for $a$ are found, but only one is given in the answer, since the original equation does not have roots for the other value.