Most of your work is correct. You used the product rule and arrived at the first derivative of -xex - ex or, if you factor out a -ex, it would be -ex(x+1).
The next step, and it looks like you tried this but got a little confused here, is to set the derivative equal to zero. By the zero product principle, that means that either: -ex = 0 or x+1 = 0.
As you noted, ex is always greater than zero, so that cannot be a solution. Therefore, the only critical point of your function is x+1 = 0 or x= -1.
Armed with this knowledge, you must now find out whether the critical point is a local maximum, a local minimum, or neither. You should be able to do that fairly easily.