Find the sum of all possible $a^3$, where $a$ is a rational figure (Heuristic Solution)

Given that $a$ is rational and the equation $ax^2+(a+2)x+a-1=0$ has integer roots.

Find the sum of all possible $a^3$.

It is very important for me to say out loud here that this solution is provided by my math friend, a retired math professor from the U.K.