1. **Stating the problem:** We are given the quadratic function $$y = a(x + h)^2 + k$$ with constants $a$, $h$, and $k$. The graph is a parabola opening downwards with vertex at $(-h, k)$ located in the bottom-right quarter of the coordinate plane. 2. **Recall the vertex form and properties:** The vertex form of a parabola is $$y = a(x - x_0)^2 + y_0$$ where $(x_0, y_0)$ is the vertex. Here, the vertex is at $(-h, k)$. 3. **Analyze the parabola opening direction:** Since the parabola opens downwards, the coefficient $a$ must be negative, i.e., $$a < 0$$. 4. **Location of the vertex:** The vertex is in the bottom-right quarter, meaning: - $x$-coordinate of vertex $-h > 0 \implies h < 0$ - $y$-coordinate of vertex $k < 0$ 5. **Evaluate each statement:** - I. $h > 0$? From step 4, $h < 0$, so I is false. - II. $k > 0$? From step 4, $k < 0$, so II is false. - III. $ah > 0$? Since $a < 0$ and $h < 0$, their product $ah > 0$ (negative times negative is positive), so III is true. 6. **Conclusion:** Only statement III is true. **Answer:** None of the options exactly match only III true, but from the given options, the only true statement is III. **Final:** The correct choice is not listed, but based on the analysis, only III is true.