The directrix of a parabola is y=−4y=−4 . the focus of the parabola is (−2,−2)(−2,−2) . what is the equation of the parabola?

Because the focus is (-2,-2) and the directrix is y = -4, the vertex is (-2,-3).

Consider an arbitrary point (x,y) on the parabola.
The square of the distance between the focus and P is
(y+2)² + (x+2)²
The square of the distance from the point to the directrix is
(y+4)²

Therefore
(y+4)² = (y+2)² + (x+2)²
y² + 8y + 16 = y² + 4y + 4 + (x+2)²
4y = (x+2)² - 12
y = (1/4)(x+2)² - 3