The correct standard form of theequation of a parabola with
a vertical axis of symmetry is

(x - h)² = 4p(y - k)

Your parabola equation, 

(x + 1)² + 8(y + 2) = 0

is not quite in that form.  So we subtract 8(y + 2)
from both sides:

           (x + 1)² = -8(y + 2)

Compare that to

           (x - h)² = 4p(y - k) 

And we see that h = -1, k = -2 and 4p = -8
                                    p = -2

We plot the vertex (h,k) = (-1,-2), and since p = -2, a
negative number, the focus is 2 units BELOW the vertex. 
So the focus is (-1,-4).  And the directrix is 2
units ABOVE the vertex and therefore has equation y = 0,
which just happens to be the x-axis.  The focal chord or
latin rectum is a line through the focus, bisected by the
focus ad which is |4p| = 8 units long.  So we draw in the latus
rectum, in green: 
 



and sketch in the parabola:



Edwin