Why Doesn't the Cover-Up Method Work for Repeated Roots?

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.

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