Final answer:
The transfer function P(s) = (s−1) / (s+2)(s²+4) represents a stable system because all the poles have negative real parts or are purely imaginary and there are no poles with a positive real part that would indicate instability.
Step-by-step explanation:
To determine which functions represent stable systems and which represent unstable systems, we must analyze the poles of the transfer function P(s) = (s−1) / (s+2)(s²+4). A system is considered stable if all poles of its transfer function have negative real parts. In this case, the denominator of the transfer function is (s+2)(s²+4). We can see that s = -2 is a pole with a negative real part, which suggests stability. However, we need to factor the quadratic equation s²+4 to find its roots. The roots of s²+4 are purely imaginary, specifically s = +2i and s = -2i, since there are no real parts to these roots, they do not make the system unstable.
Thus, since all the poles either have negative real parts or are purely imaginary, the given transfer function represents a stable system. There's no pole with a positive real part that would indicate instability.