Guess stability based on filter coefficients
In the general case there is no simple way to determine the stability of a discrete-time system (unless it is FIR, in which case it is always stable). In order to check stability you don't necessarily need to compute the poles of the transfer function, but you can use the Jury stability criterion, which does not explicitly compute the poles, because the latter is generally a numerically ill-conditioned problem.

For a second-order system the denominator coefficients have to be inside the stability triangle, which can be checked very easily.