jmc

Squaring both sides, we get $$x+\sqrt{3x+6}+2\sqrt{x+\sqrt{3x+6}}\sqrt{x-\sqrt{3x+6}}+x-\sqrt{3x+6}=36.$$Then $$2x+2\sqrt{x^2-3x-6}=36,$$so $$\sqrt{x^2-3x-6}=18-x.$$Squaring both sides, we get $x^2-3x-6=324-36x+x^2.$ Hence, $x=\boxed{10}.$ We check that this solution works.