Problem 9.1. Let $f(x)=x^{2}+(2 a-1) x-a-3$, where $a$ is a real parameter.

a) Prove that the equation $f(x)=0$ has two distinct real roots $x_{1}$ and $x_{2}$.

b) Find all values of $a$ such that $x_{1}^{3}+x_{2}^{3}=-72$.

Peter Boyvalenkov