To solve the given problem, we need to construct a polynomial $f(x)$ of degree 3 that has the specified zeros: $-2$ with multiplicity 2, and $4$. The solution should be left in factored form.

Step-by-Step Solution:

1. Understand the Zeros:

   • A zero of $-2$ with multiplicity 2 means $(x + 2)^2$ is a factor of the polynomial.
   • A zero of $4$ means $(x - 4)$ is also a factor.

2. Construct the Polynomial: Combining these factors, the polynomial can be written as: $f(x) = (x + 2)^2 (x - 4)$

3. Expand if Necessary: If expansion is required later, you can calculate: $(x + 2)^2 = x^2 + 4x + 4$ Then multiply it by $(x - 4)$:

   = x^3 - 4x^2 + 4x^2 - 16x + 4x - 16 = x^3 - 16x - 16$$

However, in factored form, the answer remains: $f(x) = (x + 2)^2 (x - 4)$

Let me know if you'd like further details! Here are some related questions to deepen understanding:

1. How do we handle polynomials with more than one zero of the same value?
2. Why does multiplicity affect the factors of a polynomial?
3. What is the difference between expanded and factored forms of a polynomial?
4. How can we confirm that the given zeros satisfy the polynomial?
5. How does the degree of a polynomial relate to the number of its roots?

Tip: Always double-check the multiplicity of zeros to ensure the polynomial matches the given conditions!