Find the sum of all complex values of $a,$ such that the polynomial $x^4 + (a^2 - 1) x^2 + a^3$ has exactly two distinct complex roots.
The polynomial must be of the form (x - r)^2*(x - s)^2. Expanding and comparing coefficients:
-2r - 2s = 0
r^2 + 4rs + s^2 = a^2 - 1
-2rs^2 - 2r^2s = 0
r^2s^2 = a^3
Solving this system, the sum of all possible values of a is 7.