#### Answer

Answer choice $H$: The graph would shift $7$ units up.

#### Work Step by Step

Let's assume that the $x$-value as 1. The $y$-value in $y = x^2-5$ is $-4$.
The $y$-value is $y = x^2 + 2$ is $3$. $3$ minus $(-4)$ is the same thing as $3+4$ which is $7$. The only answer choice that has the number $7$ in it is $\boxed{H}$