Answer
No solution.
Work Step by Step
We solve:
$\sqrt{x+2}+5=\sqrt{x+15}$
First we square both sides:
$(\sqrt{x+2}+5)^{2}=(\sqrt{x+15})^{2}$
$(\sqrt{x+2}+5)(\sqrt{x+2}+5)=(\sqrt{x+15})(\sqrt{x+15})$
$x+2+10\sqrt{x+2}+25=x+15$
$x+27+10\sqrt{x+2}=x+15$
$10\sqrt{x+2}=15-27$
$10\sqrt{x+2}=-12$
$5\sqrt{x+2}=-6$
We square both sides again:
$(5\sqrt{x+2})^{2}=(-6)^{2}$
$25(x+2)=36$
$25x+50=36$
$25x=-14$
$x=-\frac{14}{25}$
However, the solution $x=-\frac{14}{25}$ does not work in the original equation:
$\sqrt{-\frac{14}{25}+2}+5=\sqrt{-\frac{14}{25}+15}$
$\sqrt{-\frac{14}{25}+\frac{50}{25}}+5=\sqrt{-\frac{14}{25}+\frac{375}{25}}$
$\sqrt{\frac{36}{25}}+5=\sqrt{\frac{361}{25}}$
$\frac{6}{5}+\frac{25}{5}=\frac{19}{5}$
$\frac{31}{5}=\frac{19}{5}$
$31=19$
This is a false statement, so there is no solution.