1. **State the problem:** Find the equation of the parabola with vertex at $(5, -2)$ and focus at $(-5, -2)$. 2. **Recall the parabola definition:** A parabola is the set of points equidistant from the focus and the directrix. 3. **Find the vertex and focus coordinates:** Vertex $V = (5, -2)$, Focus $F = (-5, -2)$. 4. **Determine the axis of symmetry:** Since the $y$-coordinates of vertex and focus are the same, the parabola opens horizontally. 5. **Calculate the focal length $p$:** The distance from vertex to focus is $$p = |x_F - x_V| = |-5 - 5| = 10.$$ Since the focus is to the left of the vertex, the parabola opens to the left, so $p = -10$. 6. **Write the standard form of a horizontal parabola:** $$ (y - k)^2 = 4p(x - h) $$ where $(h, k)$ is the vertex. 7. **Substitute values:** $$ (y - (-2))^2 = 4 \times (-10) \times (x - 5) $$ which simplifies to $$ (y + 2)^2 = -40(x - 5). $$ 8. **Final answer:** The equation of the parabola is $$\boxed{(y + 2)^2 = -40(x - 5)}.$$