Let y = 9x + 6x^2 - 7
Thus, y' = 9 + 12x
Setting y' to zero, we have:
y' = 0
9 + 12x = 0
12x = -9
x = -9 / 12
x = -3 / 4
y'' = 12 > 0, so we have a local minimum as x = -3/4.
When x = -3/4, y = 9(-3/4) + 6(-3/4)^2 - 7 = -83 / 8
Thus, the local minimum is y = -83 / 8
Note that there are no local maximum.
This is a quadratic function, so it can only have a local max or a local min.
If you are talking about absolute maximum, that will be infinity, since the function will not stop increasing.
If you are talking about absolute minimum, that will also be y = -83/8.