The exhaustive set of real values of x for which, the function, $f(x)=\sqrt{(x+2)(5-x)}-\frac{1}{\sqrt{x^2-4}}$, is a real valued function, is

(2, 5]

The domain of $\sqrt{(x+2)(5-x)}$ is the set of values of x for which $(x+2)(5-x)≥0$

i.e., $x∈[−2, 5]$

The domain of $\frac{1}{\sqrt{x^2-4}}$ is the set of values of x for which $x^2-4>0$

i.e., $x∈(−∞, −2)∪(2, ∞)$. The common values are $(2, 5]$