A geometric sequence has first term $a\ne 0$ and common ratio $r$.

The sum of the first $4$ terms of the geometric sequence is equal to $0$.

Prove that $r=-1$ is the only possible value of $r$.

We have

Dividing by $a$ gives

Since $r=-1$ is a root, we know $r+1$ is a factor.

Try using long division to divide $r^3+r^2+r+1$ by $r+1$.