First of all, let's determine the vertex of the parabola. In general, for any vertical parabola like the one we have here, the x-coordinate of the vertex is the same as the x-coordinate of the focus. In this case, the x-coordinate of the focus is -4, so the x-coordinate of the vertex is also -4.
Next, the y-coordinate of the vertex is the midpoint between the y-coordinate of the focus and the directrix. Specifically, the formula for finding this midpoint is (y1 + y2)/2. Let's plug in our numbers: (-2 (the y-coordinate of the focus) + 3 (the y-coordinate of the directrix))/2 = 0.5. So, the vertex of the parabola is located at (-4, 0.5).
Now, let's write the equation for the parabola. For a vertical parabola, this equation is in the form: (x-h)^2 = 4p(y-k), where (h, k) is the vertex and p is the distance from the vertex to the focus (or from the vertex to the directrix), both of which are the same.
So, we need to find p, which is the absolute value of the difference between the y-coordinate of the vertex and that of the focus: abs(0.5 - (-2)) = 2.5.
Finally, let's plug everything into our parabola equation. Replacing h with -4, k with 0.5, and p with 2.5, we get: (x-(-4))^2 = 4*2.5(y-0.5). Simplifying this, we end up with the equation 'y - 0.5 = 1/10*(x - -4)^2'.
So, the equation of the parabola with a focus at (-4,-2) and a directrix at y=3 is 'y - 0.5 = 1/10*(x - -4)^2'.