Square both sides:
2x-6+2√((2x-6)(x+4))+x+4=25;
2√((2x-6)(x+4))=27-3x;
Square again:
4(2x-6)(x+4)=729-162x+9x^2;
4(2x^2+2x-24)=729-162x+9x^2;
8x^2+8x-96=729-162x+9x^2;
x^2-170x+825=0=(x-165)(x-5), so x=5 or 165.
Substitute back into the original:
2x-6=4 or 324, with square roots 2 or 18.
x+4=9 or 169 with square roots 3 or 13
When we add these we get 5 or 31, so x=5 is clearly the solution.
(Note, however, that if x=165, and we take -√(x+4) we get -13 and 18-13=5, which also fits. But square root generally implies the positive root, so x=5 would be the only solution. The ambiguity arises out of squaring twice in the course of solving the problem. x=5 or 165 applies to √(2x-6)±√(x+4)=5.)