Question

is there a value for h that makes it possible for the equation square root x+h+5= 0 to have any real number solutions

Answer 1

This equation? \[
\sqrt{x+h+5} = 0
\]

Answer 2

Yes.
for (x+h+5)^1/2
to be real
x+h+5 has to be postie. Otherwise you're dealing with imaginary numbers.
Therefore
x+h+5>=0
x+h>-5