Trick to this square root equations

write like this

$$\sqrt{x+14}+\sqrt{x-7}=\sqrt{x-2}+\sqrt{x+5}$$

square both sides:

$$2x+7+2\sqrt{(x+14)(x-7)}=2x+3+2\sqrt{(x-2)(x+5)}\\ 2+\sqrt{(x+14)(x-7)}=\sqrt{(x-2)(x+5)}$$

square again

$$4\sqrt{(x+14)(x-7)}+x^2+7x-94=x^2+3x-10\\ \sqrt{(x+14)(x-7)}=21-x \to (x+14)(x-7)=(21-x)^2$$

Can you finish?

Don't forget to test the result in the original equation.