y = -2x2 + 8x - 15
Move the '-15' term to the other side by adding 15 to both sides:
y + 15 = -2x2 + 8x
Factor out the '-2' term:
y + 15 = -2(x2 - 4x)
Complete the square of x2 - 4x ---> x2 - 4x + 4
Notice that the '4' will be on the inside of the factor, so really, it is a '-8':
y = 15 - 8 = -2(x2 - 4x + 4)
Finishing:
y + 7 = -2(x - 2)2
The vertex occurs at (2, -7)