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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.

Use the graph of the polynomial function h(x) to determine where h(x) < 0 and where h(x) = 0.

Learn how to analyze the graph of a polynomial function to determine where the function is negative and where it intersects the x-axis. This problem covers key algebraic concepts, including polynomial function analysis and interval identification. Suitable for students in Grades 10-12.

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