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To solve this problem, we need to construct a polynomial \( f(x) \) with the given degree and zeros. 

### Given Information:
- Degree: 4
- Zeros: \( 4 \), \( -5i \), and \( 5i \) with multiplicity 2.

### Step-by-Step Solution:
1. **Understanding the Roots**:
   - Since the polynomial has real coefficients, the complex zeros must come in conjugate pairs.
   - The given zeros are: \( 4 \), \( -5i \), and \( 5i \) (since complex conjugates are required).
   - There is a multiplicity of 2 for \( -5i \) and \( 5i \).

2. **Forming the Factors**:
   - The factor corresponding to the zero \( 4 \) is \( (x - 4) \).
   - The factors for the zeros \( -5i \) and \( 5i \) can be expressed as:
     \[
     (x - (-5i))(x - 5i) = (x + 5i)(x - 5i) = x^2 + 25
     \]
   - Since there is a multiplicity of 2 for these zeros, we multiply this factor twice, resulting in:
     \[
     (x^2 + 25)^2
     \]

3. **Forming the Polynomial**:
   - The polynomial will be the product of these factors. Let \( a \) be the leading coefficient, so:
     \[
     f(x) = a(x - 4)(x^2 + 25)^2
     \]

Thus, the polynomial is \( f(x) = a(x - 4)(x^2 + 25)^2 \).

Would you like further details, or do you have any questions?

Here are 5 follow-up questions:
1. How do you expand \( (x^2 + 25)^2 \)?
2. How does the leading coefficient \( a \) affect the shape of the polynomial?
3. Why do complex conjugate pairs need to exist for polynomials with real coefficients?
4. How would the polynomial change if the zero \( 4 \) had a different multiplicity?
5. What is the significance of a root’s multiplicity in a polynomial's graph?

**Tip**: When working with complex roots in real polynomials, always remember to include their conjugates!

Form a polynomial f(x) with real coefficients having the given degree and zeros. Degree 4; zeros: 4; -5i; 5i multiplicity 2.

This problem demonstrates how to construct a degree-4 polynomial with real coefficients given specific zeros, including complex conjugates and multiplicities. The solution includes forming factors for real and complex roots.

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