The correct answer is: A. It is a finite duration signal.
A causal signal is a signal whose value at time $n$ depends only on values of the input signal at times $n \leq 0$. A non-causal signal is a signal whose value at time $n$ depends on values of the input signal at times $n < 0$. A periodic signal is a signal that repeats itself at regular intervals.
The z-transform of a causal signal is a rational function with no poles in the region of convergence (ROC) outside the unit circle. The z-transform of a non-causal signal is a rational function with at least one pole in the region of convergence outside the unit circle. The z-transform of a periodic signal is a rational function with at least one pole on the unit circle.
In this case, the z-transform of the signal $x[n]$ is a rational function with two poles at $z = \pm 2j$. These poles are inside the unit circle, so the ROC of the z-transform is the entire complex plane. This means that the signal $x[n]$ is a finite duration signal.