Sander and andrew2322 are using the fact that |z- a| is the distance from the point z to the point a in the complex plane. |z+ 5|= |z- (-5)| so |z+ 5|= 4 is satisfied for all points whose distanced from -5 is equal to 4- in other words, the circle with center -5 and radius 4.
Also, you arrived at
and then proceeded to multiply out the square, etc. You should not have done that. You should have recognized immediately that this is of the form
, the equation of a circle with center (a, b) and radius r, with a= -5, b= 0, and r= 4.