3x + y^2 + 8y + 4 = 0

3x + 4 = -(y^2 + 8y)

Now complete the square:

3x + 4 = -(y^2 + 8y + 16 - 16)

3x + 4 = -(y^2 + 8y + 16) + 16

3x - 12 = -(y + 4)^2

Now factor out a 3.

3(x - 4) = -(y + 4)^2

Now write this as

4*(3/4)(x - 4) = -(y + 4)^2

Compare this to the form of a leftward opening parabola with a vertex at (h, k):

4p(x - h) = -(y - k)^2

where p is the distance from the vertex to the focus, and the distance from the focus to the directrix.

So p = 3/4, and the vertex is at (4, -4).

The focus is a point p = 3/4 units to the left of the vertex:

(4 - 3/4, -4) = (13/4, -4)

And the directrix is a vertical line p = 3/4 units to the right of this point:

x = 4 + 3/4 = 19/4

-Dan