https://file.aiquickdraw.com/mathbot/file/17307818328691xxubwiy.jpg

To sketch a polynomial \( f(x) \) with the given characteristics, let’s analyze each condition:

### 1. Polynomial Degree
- The polynomial has a degree of 4, which means the leading term will be \( x^4 \), making it a quartic polynomial.

### 2. Roots
- The polynomial has roots at:
  - \( x = -4 \) (multiplicity 1, so it crosses the x-axis at this root).
  - \( x = -2 \) (multiplicity 1, so it also crosses the x-axis here).
  - \( x = 3 \) (multiplicity 2, so it touches the x-axis and does not cross it at this root).

### 3. End Behavior
- As \( x \to -\infty \), \( f(x) \to -\infty \).
- As \( x \to \infty \), \( f(x) \to -\infty \).
  
Since the end behavior of the polynomial shows that \( f(x) \) approaches \( -\infty \) on both ends, the leading coefficient must be negative. Thus, the polynomial will have the form:
\[
f(x) = -a(x + 4)(x + 2)(x - 3)^2
\]
where \( a > 0 \) (to ensure the correct end behavior).

### Steps to Sketch
1. **Plot the roots** on the x-axis at \( x = -4 \), \( x = -2 \), and \( x = 3 \).
2. **Behavior at each root**:
   - At \( x = -4 \) and \( x = -2 \): the function crosses the x-axis.
   - At \( x = 3 \): the function touches the x-axis and turns back, due to the multiplicity of 2.
3. **End Behavior**: The graph heads down to \( -\infty \) as \( x \) approaches both \( -\infty \) and \( +\infty \).

Would you like a specific graph of this polynomial, or further explanation on any of the steps?

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### Related Questions:
1. How does the multiplicity of a root affect the shape of the graph near that root?
2. Why does a negative leading coefficient in a quartic polynomial result in both ends going to \( -\infty \)?
3. What happens if we change the multiplicity of the root at \( x = 3 \) to 1?
4. How would the graph look if the leading coefficient were positive?
5. Can a quartic polynomial have end behavior where one end goes to \( +\infty \) and the other to \( -\infty \)?

### Tip:
When sketching polynomials, always check the end behavior first, as it gives you a good idea of the overall direction of the graph on both sides.

Sketch a polynomial that has these characteristics:
- It has degree 4
- It has roots at -4, -2, and 3 (multiplicity 2)
- As x approaches -∞, f(x) approaches -∞
- As x approaches ∞, f(x) approaches -∞

Explore how to sketch a degree 4 polynomial with roots at -4, -2, and 3 (with a root at 3 having multiplicity 2). Understand polynomial end behavior and how the leading coefficient affects the shape of the graph.

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