Write the equation of a parabola with a vertex at (-5, 2) and a directrix y = -1.

The vertex is halfway between focus and directrix, so we know the focus is at (-5,5).
Points on the parabola have the form (x, f(x)). They all have equal distance to the focus and the directrix, so they satisfy the following equation:
f(x) - (-1) = sqrt( ( x - (-5) )^2 + (f(x) - 5)^2)
(Left side is distance to directrix, only depends on the y-coordinate of the point. Right side is the distance between point an focus computed using the pythoagorean theorem.)
Squaring both sides  and collecting the minuses gives
(f(x) +1)^2 = ( x + 5 )^2 + (f(x) - 5)^2
Multiply out the brackets using the binomial theorem:
f(x)^2 + 2*f(x) + 1 = x^2 + 10x + 25  + f(x)^2 - 10f(x) + 25
Simplify by subtracting f(x)^2. Then bring +1 to the right side and -10f(x) to the left side. This gives
12*f(x) = x^2 + 10 x + 49
Divide by 12 and you get the final function:
f(x) = 1/12 *x^2 + 10/12 * x + 49/12