To find the roots of the polynomial function [tex]\( f(x) = \left(x^2 + 2x - 15\right)\left(x^2 + 8x + 17\right) \)[/tex], we need to solve for the values of [tex]\( x \)[/tex] that make [tex]\( f(x) = 0 \)[/tex].
Let's break down the factors individually:
1. The first polynomial factor is [tex]\( x^2 + 2x - 15 \)[/tex]. We can factor this quadratic expression to find its roots:
[tex]\[
x^2 + 2x - 15 = (x + 5)(x - 3)
\][/tex]
Setting each factor equal to zero gives us:
[tex]\[
x + 5 = 0 \quad \Rightarrow \quad x = -5
\][/tex]
[tex]\[
x - 3 = 0 \quad \Rightarrow \quad x = 3
\][/tex]
Therefore, the roots of the first polynomial factor are [tex]\( x = -5 \)[/tex] and [tex]\( x = 3 \)[/tex].
2. The second polynomial factor is [tex]\( x^2 + 8x + 17 \)[/tex]. To find its roots, we use the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = 8 \)[/tex], and [tex]\( c = 17 \)[/tex]:
[tex]\[
x = \frac{-8 \pm \sqrt{8^2 - 4 \cdot 1 \cdot 17}}{2 \cdot 1}
\][/tex]
[tex]\[
x = \frac{-8 \pm \sqrt{64 - 68}}{2}
\][/tex]
[tex]\[
x = \frac{-8 \pm \sqrt{-4}}{2}
\][/tex]
Since [tex]\(\sqrt{-4} = 2i\)[/tex]:
[tex]\[
x = \frac{-8 \pm 2i}{2}
\][/tex]
Simplifying this, we get:
[tex]\[
x = -4 \pm i
\][/tex]
Therefore, the roots of the second polynomial factor are [tex]\( x = -4 + i \)[/tex] and [tex]\( x = -4 - i \)[/tex].
Combining both sets of roots, the complete list of roots for the polynomial [tex]\( f(x) = \left(x^2 + 2x - 15\right)\left(x^2 + 8x + 17\right) \)[/tex] is:
[tex]\[
x = -5, \quad 3, \quad -4 + i, \quad -4 - i
\][/tex]
Among the given options, the correct complete list of roots is:
[tex]\[
-5, \; 3, \; -4 + i, \; -4 - i
\][/tex]
