Answer
The equation has no solutions.
Work Step by Step
Square both sides
$(\sqrt{x+5}+2)^{2}=(\sqrt{x-1})^{2}$
$ x+4\sqrt{x+5}+9=x-1\qquad$ ...Add $-x-9$ to both sides
$4\sqrt{x+5}=-10 \qquad$ ... Square both sides
$16(x+5)=100$
$16x+80=100 \qquad$ ... Subtract 80 from both sides
$16x+80-80=100-80$
$ 16x=20\qquad$ ...Divide both sides by 16
$x=\displaystyle \frac{5}{4}$
Verify Solution:
LHS: $\displaystyle \sqrt{\left(\frac{5}{4}\right)+5}+2=\frac{9}{2}$
RHS: $\displaystyle \sqrt{\left(\frac{5}{4}\right)-1}=\frac{1}{2}$
$x=\displaystyle \frac{5}{4}$ is not a solution. The equation has no solutions.