y = (1/4)(x^2 + 5x - 1)

Complete the square inside the parenthesis:

y = (1/4)([x^2 + 5x + (5/2)^2] - (5/2) - 1)

y = (1/4)(x + 5/2)^2 - (1/4)(7/2)

y = (1/4)(x + 5/2)^2 - 7/8

4(y + 7/32) = (x + 5/2)^2

The general form for an upward opening parabola with a vertex at (h, k) is:

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

where p is the distance of the vertex to the focus, or the distance of the vertex to the directrix.

So the vertex of your parabola is

(h, k) = (-5/2, -7/32)

The focus is p units directly above the vertex, and p = 1 (according to the above form) so the focus is at

(-5/2, -7/32 + 1) = (-5/2, 25/32)

The directrix is, in this case, a horizontal line p = 1 units below the vertex, so this line is

y = -7/32 - 1 = -39/32

-Dan