Answer
No solution.
Work Step by Step
We are given:
$\sqrt{x+5}+2=\sqrt{x-1}$
We square both sides:
$(\sqrt{x+5}+2)^{2}=(\sqrt{x-1})^{2}$
$(\sqrt{x+5}+2)(\sqrt{x+5}+2)=(\sqrt{x-1})(\sqrt{x-1})$
And distribute:
$(x+5)+4\sqrt{x+5}+4=x-1$
And combine like terms:
$9+4\sqrt{x+5}+1=x-x$
$4\sqrt{x+5}=-10$
$2\sqrt{x+5}=-5$
We square both sides again:
$(2\sqrt{x+5})^{2}=(-5)^{2}$
And distribute:
$4(x+5)=25$
$4x+20=25$
$4x=5$
$x=\frac{5}{4}$
However, the solution $\frac{5}{4}$ does not work in the original equation:
$\sqrt{\frac{5}{4}+5}+2=\sqrt{\frac{5}{4}-1}$
$\sqrt{\frac{5}{4}+\frac{20}{4}}+2=\sqrt{\frac{5}{4}-\frac{4}{4}}$
$\sqrt{\frac{25}{4}}+2=\sqrt{\frac{1}{4}}$
$\frac{\sqrt{25}}{\sqrt{4}}+2=\frac{\sqrt{1}}{\sqrt{4}}$
$\frac{5}{2}+2=\frac{1}{2}$
$\frac{9}{2}=\frac{1}{2}$
$9=1$
Since we got a false statement, the solution $\frac{5}{4}$ does not work. Hence there is no solution.