Let's analyze the problems step-by-step.

Problem 2: $f(x) = x(x - 1)(x + 2)$

1. X-intercepts: The x-intercepts are determined by setting $f(x) = 0$:

   • $x = 0$
   • $x - 1 = 0$ → $x = 1$
   • $x + 2 = 0$ → $x = -2$

2. Behavior at the x-intercepts:

   • At $x = 0$: The graph crosses the x-axis because the factor $x$ has an odd multiplicity (1).
   • At $x = 1$: The graph crosses the x-axis because the factor $x - 1$ has an odd multiplicity (1).
   • At $x = -2$: The graph crosses the x-axis because the factor $x + 2$ has an odd multiplicity (1).

3. Number of turning points: A cubic function has at most $n - 1 = 2$ turning points. By graphing, the function shows two turning points.

4. Number and type of extrema:

   • There is one local maximum and one local minimum. These are not global extrema because the cubic function extends infinitely.

Problem 3: $f(x) = -(x + 5)(x + 1)^2(x - 2)$

1. X-intercepts: The x-intercepts are determined by setting $f(x) = 0$:

   • $x + 5 = 0$ → $x = -5$
   • $x + 1 = 0$ → $x = -1$ (with multiplicity 2)
   • $x - 2 = 0$ → $x = 2$

2. Behavior at the x-intercepts:

   • At $x = -5$: The graph crosses the x-axis because the factor $x + 5$ has an odd multiplicity (1).
   • At $x = -1$: The graph is tangent to the x-axis because the factor $(x + 1)^2$ has an even multiplicity (2).
   • At $x = 2$: The graph crosses the x-axis because the factor $x - 2$ has an odd multiplicity (1).

3. Number of turning points: A quartic function has at most $n - 1 = 3$ turning points. By graphing, the function shows three turning points.

4. Number and type of extrema:

   • There is one local maximum and two local minima. These are not global extrema because the quartic function extends infinitely.

Problem 4: Sketch $f(x) = x(x - 2)(x + 1)$

This is another cubic function.

1. X-intercepts:
   • $x = 0$, $x = 2$, and $x = -1$.
2. Behavior: The graph crosses the x-axis at each intercept (all have odd multiplicities).
3. General shape: As a cubic function with a positive leading coefficient, the graph falls to the left and rises to the right.

Let me know if you'd like me to graph these functions or need detailed explanations of specific parts.

Expansion Questions:

1. How can you confirm turning points algebraically without graphing?
2. What is the significance of multiplicities for x-intercepts in polynomial graphs?
3. How can end behavior of polynomial functions be predicted from the leading term?
4. What are the differences between local and global extrema in these graphs?
5. How would these graphs change if the coefficients were altered?

Tip: Use derivative tests to confirm the nature of turning points and extrema for better precision.