which of the following is the complete list of roots for the polynomial function f(x)=(x^2 +6x+8)(x^2+6+13)

-3+2i, -3-2i
-2, -4
-2, -4, -3+2i, -3-2i
-2, -4, -3+2i, 3+2i

We are given the polynomial function \( f(x) = (x^2 + 6x + 8)(x^2 + 6x + 13) \), and asked to find the complete list of roots.

### Step 1: Solve for roots of \( x^2 + 6x + 8 \)
We can factor this quadratic expression:
\[
x^2 + 6x + 8 = (x + 2)(x + 4)
\]
Thus, the roots are \( x = -2 \) and \( x = -4 \).

### Step 2: Solve for roots of \( x^2 + 6x + 13 \)
For this quadratic, the discriminant (Δ) is:
\[
\Delta = b^2 - 4ac = 6^2 - 4(1)(13) = 36 - 52 = -16
\]
Since the discriminant is negative, the roots are complex. Using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{-6 \pm \sqrt{-16}}{2} = \frac{-6 \pm 4i}{2}
\]
\[
x = -3 \pm 2i
\]
Thus, the roots are \( x = -3 + 2i \) and \( x = -3 - 2i \).

### Conclusion
The complete list of roots for the polynomial is:
\[
x = -2, -4, -3 + 2i, -3 - 2i
\]
Therefore, the correct answer is:
\[
\boxed{-2, -4, -3+2i, -3-2i}
\]

Would you like further details or have any questions? Here are five related questions to expand on this problem:

1. How do you find the discriminant of a quadratic equation?
2. What do complex roots signify in the context of polynomials?
3. Can a polynomial of degree 4 have more than four roots?
4. How does the quadratic formula change when dealing with complex numbers?
5. What is the significance of factoring polynomials in solving for their roots?

**Tip:** When the discriminant of a quadratic is negative, expect complex (imaginary) roots.

which of the following is the complete list of roots for the polynomial function f(x)=(x^2 +6x+8)(x^2+6x+13)
-3+2i, -3-2i
-2, -4
-2, -4, -3+2i, -3-2i
-2, -4, -3+2i, 3+2i

Discover how to find the complete list of roots for the polynomial function f(x) = (x^2 + 6x + 8)(x^2 + 6x + 13). This example covers factoring quadratic expressions, using the quadratic formula, and dealing with complex numbers. Ideal for high school students in Grades 10-12.

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