There's a mistake going from line 2 to 3 on the right-hand side. The x - (-5) becomes x + 5, but when you distribute the -1, you get -x - 5.
I find that there's an easier way to do these sorts of problems:
We know the slope is -1 so let's pretend the equation of the line is y = -x. We want it to run through (-5, 2), so we plug in -5 for x. y = 5 then. So we are 3 units above our goal, which is 2. So we need to come down 3. Thus, the equation is actually y = -x - 3.