The table in part b) provides possible values for the leading coefficient a of the polynomial function $m(x)$ in its factored form. Based on the graph shown in the figure, you're supposed to determine which value of a would result in the polynomial matching the behavior of the graph.

Steps to Analyze:

1. The given graph is a cubic polynomial because it crosses the x-axis three times.

2. The factored form of the polynomial given is $a(x + 2)^2(x - 3)$, which means:

   • There's a double root at $x = -2$ (since $(x + 2)^2$).
   • A single root at $x = 3$ (since $(x - 3)$).

3. Now, we need to pick the right value of a so that the graph of $m(x)$ matches the provided graph.

   • If a is positive, the end behavior of the cubic function would go from $-\infty$ to $+\infty$, meaning the graph rises to the right.
   • If a is negative, the end behavior would be reversed (from $+\infty$ to $-\infty$), meaning the graph falls to the right.

Analysis of the Graph:

• The graph in the image falls to the right, meaning the leading coefficient a must be negative.

Conclusion:

From the possible values in the table (0.3, 0.2, -0.2, -0.3), the correct value for a would likely be -0.3, since the graph has steep behavior and requires a stronger negative coefficient to match the steepness and shape of the graph.

Do you have any questions or need more details?

Here are 5 related questions for further exploration:

1. What is the general form of a cubic polynomial?
2. How do double roots affect the shape of the graph of a polynomial?
3. Why does the leading coefficient affect the end behavior of a polynomial graph?
4. How can you determine the multiplicity of roots from a graph?
5. What role does the degree of a polynomial play in its graph?

Tip: Always observe the direction of the graph as it moves towards infinity to identify the sign of the leading coefficient.