Given that one of the roots is $4 + i$, we must remember that the coefficients of polynomials with real numbers are real. This means that if a polynomial has a non-real (complex) root, its complex conjugate must also be a root. Therefore, the conjugate of $4 + i$ is $4 - i$, and it must also be a root.

Now, the polynomial is of degree 5, and we already know three of its roots: $-2$, $2$, and $4 + i$. The conjugate $4 - i$ will add another root, giving us 4 roots so far.

Since the degree of the polynomial is 5, there must be one more root. This root must be a real number because complex roots occur in conjugate pairs, and we only need one more root.

Summary of the roots:

• 2 real roots: $-2$, $2$
• 2 complex roots: $4 + i$ and $4 - i$
• 1 remaining real root.

Therefore, the statement that best describes the number and nature of all roots is:

f of x has 3 real roots and 2 imaginary roots.

Would you like more details on this explanation or have any further questions?

Here are 5 related questions you might consider:

1. What are the general properties of the roots of polynomials with real coefficients?
2. How can you factor a polynomial if you know its complex roots?
3. What is the significance of the degree of a polynomial function in relation to its roots?
4. Can a polynomial of degree 5 have all real roots?
5. How are the imaginary roots represented graphically on the complex plane?

Tip: Always remember that the roots of polynomials with real coefficients come in complex conjugate pairs if they are non-real.