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!