Good point, well made.
Clearly, differentiating my answers showed that only the +ve solution holds true (also looking over it again, I realise I dropped my + C).
But I can not see any reason why if u^2 = x + 2, then u only equals +sqrt(x+2)
After all, if you were to graph u^2 = x + 2 then you have a complete side ways parabola with its vertex at (0 , -2).
So unless there is a reason why we can ignore the -ve value of u (and I am beginning to think there might be, but apart from the final answer not working I can not see it), then to be mathematically correct you would have to give both +ve and -ve answers but then state that only one of your answers actually fits. It would be like an absolute value question where you must test your answers because sometimes one or more values do not actually fit when you sub them back in.
Also, as an aside. If the question did not state which substitution to use, then I would have just let u = +sqrt(x+2) and that would avoid this situation completely.