The graph provided shows a polynomial function $h(x)$, and you are asked to determine the intervals where:

1. $h(x) < 0$ (where the graph is below the x-axis).
2. $h(x) = 0$ (where the graph intersects the x-axis).

3a. $h(x) < 0$

To find where $h(x)$ is negative, look at the regions where the graph is below the x-axis. From the graph, the polynomial is below the x-axis between the x-intercepts:

• Between $x = -4$ and $x = -1$.
• Between $x = 1$ and $x = 3$.

Thus, $h(x) < 0$ on the intervals: $(-4, -1) \cup (1, 3)$

3b. $h(x) = 0$

To find where $h(x) = 0$, identify the points where the graph intersects the x-axis. From the graph, this happens at:

$x = -4, -1, 1, 3$

So, $h(x) = 0$ at: $x = -4, -1, 1, 3$

Would you like more details or further clarification?

Here are five related questions:

1. What does the behavior of the graph near the x-axis suggest about the multiplicity of the roots?
2. How can we determine the degree of the polynomial from this graph?
3. What is the significance of the intervals where $h(x) > 0$?
4. How can we use the graph to identify local maxima and minima?
5. How would this analysis change if there were more turning points?

Tip: When analyzing a graph of a polynomial, always check for where it crosses the x-axis (roots) and whether it touches or crosses through, as this indicates the nature of the roots.