Answer
$x^{2}+4$ is prime. Check proof below.
Work Step by Step
Recall that quadratic functions come in the form $ax^{2}+bx+c$
Notice that $x^{2}+4$ is the same as $x^{2}+0x+4$
First we list the pairs of integers that multiply to 4, along with their sum:
1 and 4 -> 5
-1 and -4 -> -5
2 and 2 -> 4
-2 and -2 -> -4
None of these sums are equal to 0, therefore $x^{2}+4$ is prime
