OK, first note that the point (-2, 2) is not on the parabola x<sup>2</sup> = 8y, so there is an error in your question. You might mean: Find the equation of the tangent to the parabola x<sup>2</sup> = 8y at the point where x = -2, in which case the answer is:
Making y the subject of x<sup>2</sup> = 8y, we have y = x<sup>2</sup> / 8
dy/dx = 2x / 8 = x / 4
At x = -2, m<sub>tang</sub> = -2 / 4 = -1 / 2
And, y = (-2)<sup>2</sup> / 8 = 4 / 8 = 1 / 2.
Using the point gradient form of a line, we need the line through (-2, 1 / 2) with gradient -1 / 2. This is:
y - 1 / 2 = (-1 / 2) * (x - -2), which becomes, on multiplication by 2,
2y - 1 = -(x + 2)
ie. x + 2y + 1 = 0, in general form.
Alternately, you may have meant "Find the equation(s) of all tangents to the parabola x<sup>2</sup> = 8y that pass through the point (-2, 2)." This is a harder question.
Let the equation(s) of all such tangent(s) have the form y = mx + b. We know that these tangent(s) pass through (-2, 2), and so:
2 = m(-2) + b
b = 2 + 2m
So, the tangent(s) have the form y = mx + 2m + 2.
These tangent(s) meet the parabola x<sup>2</sup> = 8y, or y = x<sup>2</sup> / 8, when
x<sup>2</sup> / 8 = mx + 2m + 2
x<sup>2</sup> - 8mx - 8(2m + 2) = 0
Since a tangent meets a parabola at only one point, this equation must have only one solution, and so its discriminant is zero.
Thus, (-8m)<sup>2</sup> - 4(1)[-8(2m + 2)] = 0
64m<sup>2</sup> + 64m + 64 = 0
64(m<sup>2</sup> + m + 1) = 0
This equation has no real solution, and so the answer would be that there is no tangent to the parabola x<sup>2</sup> = 8y that passes through (-2, 2).
The fact that there is no answer leads me to suspect that my first interpretation is correct, although other interpretations are possible. For example, you might have meant the point (-4, 2), or the parabola x<sup>2</sup> = 2y, or ...
