To analyze the given image and answer the questions:

1. Question (a): Intervals where the function is decreasing

   • From the graph, the function appears to be decreasing in the following intervals:
     • $(-\infty, -6)$
     • $(-3, 2)$
     • $(6, 9)$

   So, the correct options to select are:

   • $(-\infty, -6)$
   • $(-3, 2)$
   • $(6, 9)$

2. Question (b): Values of $x$ where the function has local maxima

   • Observing the graph, the local maxima appear at $x = -3$ and $x = 6$.
   • Therefore, the answer is:
     • $x = -3, 6$

3. Question (c): Sign of the leading coefficient

   • The graph shows that the function's ends go in opposite directions, with the left end going upwards and the right end going downwards. This behavior suggests a negative leading coefficient (typical of an odd-degree polynomial with a negative leading term).
   • The answer is:
     • Negative

4. Question (d): Possible degree of the function

   • Counting the turning points, we observe four turning points, suggesting a degree of at least 5 or higher.
   • Given this, the possible degrees to check are:
     • 5, 7, and 9 (as these options allow for multiple turning points).

Would you like further details on any of these answers, or do you have additional questions?

Here are five related questions for deeper understanding:

1. Why does a negative leading coefficient cause the function's right end to go downwards?
2. How do turning points relate to the degree of a polynomial function?
3. What is the significance of local maxima and minima in analyzing the behavior of functions?
4. Can we determine the exact polynomial function from this graph alone?
5. What are other graphical features that might indicate the degree of a polynomial?

Tip: When analyzing polynomial functions, remember that the maximum number of turning points is one less than the degree of the polynomial.