The axis of symmetry in a quadratic equation is given by the equation x = h, where h is the value from the standard form of the quadratic equation y = a(x-h)² + k.
To find the equation of the axis of symmetry, we first review the quadratic function provided: y=-4(x+6)² -7.
We note that the quadratic function y = -4(x+6)^2 - 7 is already in the vertex form y = a(x-h)² + k. Therefore, whatever the value of h is, that is going to be the axis of symmetry.
From our equation, we see that h = -6.
Thus, the equation for the axis of symmetry for the given quadratic function y = -4(x+6)² -7 is x = -6, so the correct answer is A) X=-6.