Could use SolveAlways
to eliminate s
and solve for the rest.
SolveAlways[(s^2 + 3)/((s - 1)^2 (s + 1)) ==
A1/(s - 1) + A2/(s - 1)^2 + A3/(s + 1), s]
(* Out[9]= {{A2 -> 2, A3 -> 1, A1 -> 0}} *)
Alternatively could get the PFD directly with Apart
.
Apart[(s^2 + 3)/((s - 1)^2 (s + 1)), s]
(* Out[8]= 2/(-1 + s)^2 + 1/(1 + s) *)