√(5x+15)-√(x+7)=2, square both sides:
5x+15-2√(5x2+50x+105)+x+7=4,
6x+22-2√(5x2+50x+105)=4,
6x+18=2√(5x2+50x+105),
3x+9=√(5x2+50x+105), square again:
9x2+54x+81=5x2+50x+105,
4x2+4x-24=0,
x2+x-6=(x+3)(x-2).
There would appear to be two solutions for x: -3 and 2, but each has to be tested in the original equation.
5x+15=0 or 25; x+7=4 or 9. The square roots are 0 or 5, and 2 or 3.
The only corresponding pair with a difference of 2 is 5-3=2, so x=2 (√(5x+15)=5, √(x+7)=3) is the solution.