The image asks to graph the polynomial function $f(x) = (x + 4)^2(1 - x)$ using parts (a) through (e).

To solve and graph this polynomial function, we need to analyze it step by step:

Step 1: Find the Zeros of the Function

To find the zeros of the function $f(x) = (x + 4)^2(1 - x)$, set $f(x) = 0$.

$(x + 4)^2(1 - x) = 0$

This equation is satisfied when any of the factors equals zero:

• $(x + 4)^2 = 0$ gives $x = -4$
• $1 - x = 0$ gives $x = 1$

So, the zeros of the function are $x = -4$ (with multiplicity 2) and $x = 1$ (with multiplicity 1).

Step 2: Determine the Behavior at Each Zero

• At $x = -4$: Since the multiplicity is 2, the graph touches the x-axis at this point and turns around.
• At $x = 1$: Since the multiplicity is 1, the graph crosses the x-axis at this point.

Step 3: Determine the End Behavior of the Polynomial

The degree of the polynomial is 3 (from multiplying the factors), and the leading coefficient comes from the term $-x^3$. Since the leading term is negative and the degree is odd, the end behavior will be:

• As $x \to -\infty$, $f(x) \to \infty$
• As $x \to \infty$, $f(x) \to -\infty$

Step 4: Identify Critical Points and Intervals of Increase/Decrease

Differentiate $f(x)$ to find the critical points: $f(x) = (x + 4)^2(1 - x)$ Using the product rule: $f'(x) = 2(x + 4)(1 - x) - (x + 4)^2$ Set $f'(x) = 0$ to find the critical points and determine intervals of increase and decrease.

Step 5: Graph the Function

With the zeros, end behavior, and critical points identified, you can now graph the function.

Would you like to see the detailed derivative steps or a sketch of the graph? Here are five follow-up questions:

1. Would you like to verify the calculations of the critical points?
2. Are you interested in the step-by-step differentiation process?
3. Do you need more information on how the graph behaves between the zeros?
4. Would you like to understand the multiplicity's effect on the graph shape?
5. Are you curious about the exact intervals where the function increases or decreases?

Tip: Always analyze the multiplicity of the roots when graphing polynomials to predict whether the graph will touch or cross the x-axis.