Answer
See the details below.
Work Step by Step
We see that the polynomial $x^2+4$ is a sum of two squares and there is no factoring rule to apply for such a case. Moreover, we note that the factors of $4$ are $\pm 1,\pm 2, \pm 4$. We would have to find among them two factors such that the sum is zero (which is the middle term $0x$) and the product is $4$. There are no such factors, so we see that the expression is prime.
