I do not know how to approach this problem. It seems that the LHS is very similar to the expansion of (a - b)^2, in this case, a = z + 1/z and b = z^2 + 1/z^2. I think this has something to do with roots of unity since z and 1/z are conjugates if z is a root of unity. I know there are 14 solutions, but I only have 2: the third roots of unity excluding 1. What strategy should I use to approach this problem? Thank you.