Write the equation of the parabola y=x^2 + 8x - 51

You already have the equation! If you want it in vertex form, do this:

y = x^2 + 8x + 16 - 16 - 51, or y = x^2 + 8x + 16 - 67, or

y = (x+4) ^2 - 67.

The vertex is at (-4,67).