1. **State the problem:** We are given the parabola equation $$(x + 4)^2 = -4(y + 7)$$ and need to find the vertex, focus, focal diameter, and directrix. 2. **Identify the form:** This is a parabola in the form $$(x - h)^2 = 4p(y - k)$$ where the vertex is at $$(h, k)$$ and it opens up if $p > 0$ or down if $p < 0$. 3. **Find the vertex:** Comparing, we have $$h = -4$$ and $$k = -7$$ so the vertex is $$(-4, -7)$$. 4. **Find $p$:** The equation is $$(x + 4)^2 = -4(y + 7)$$ which matches $$(x - (-4))^2 = 4p(y - (-7))$$ so $$4p = -4$$ giving $$p = -1$$. 5. **Interpret $p$:** Since $$p = -1$$, the parabola opens downward. 6. **Find the focus:** The focus is at $$(h, k + p) = (-4, -7 - 1) = (-4, -8)$$. 7. **Find the directrix:** The directrix is the line $$y = k - p = -7 - (-1) = -6$$. 8. **Find the focal diameter:** The focal diameter is $$|4p| = |-4| = 4$$. **Final answers:** - Vertex: $$(-4, -7)$$ - Focus: $$(-4, -8)$$ - Directrix: $$y = -6$$ - Focal diameter: $$4$$