Consider the case in which one of the roots is repeated:
The expansion for this case is:
Most of the unknown coefficients can be found using the "cover-up" method.
However if we try to apply the same technique to find A2, we find that it fails:
If we evaluate this expression at s=-2, then the first and last terms drop out, but both middle terms survive. In fact the third term is undefined (a divide by 0 condition). Clearly we need another technique to find the coefficient A2.