Answer
$\{\pm 2\}$
Work Step by Step
Step 1. Rewrite the equation as $\sqrt {2x+5}=\sqrt {x+2}+1$
Step 2. Take square on both sides to get: $2x+5=x+2+2\sqrt {x+2}+1$ or $x+2=2\sqrt {x+2}$
Step 3. Take square on both sides again to get: $x^2+4x+4=4x+8$ or $x^2=4$ which gives $x=\pm 2$
Step 4. Check answers in the original equation: they both fit.