This is a min-max problem, right?
The simplest way to do this problem is to graph it. Note that your first derivative does not exist at the points x = -4 and x = -1. However note that it does exist for any value in between x = -4 and x = -1. So there are not two solutions to this problem, but a whole interval on the x axis. (I'm not sure of where you got the "correct" answers, but they are not correct.)