Which two values of $x$ are roots of the polynomial below? \[ x^{2}+5 x+9 \] A. $x=\frac{5+\sqrt{17}}{2}$ B. $x=\frac{-5-\sqrt{61}}{2}$ C. $x=\frac{-5-\sqrt{-11}}{2}$ D. $x=\frac{-5+\sqrt{61}}{2}$ E. $x=\frac{5-\sqrt{17}}{2}$ F. $x=\frac{-5+\sqrt{-11}}{2}$

Step 1 ：Given the polynomial equation \(x^{2}+5x+9=0\), we are asked to find the roots of this equation.

Step 2 ：The roots of a quadratic equation \(ax^2 + bx + c = 0\) can be found using the quadratic formula \(\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

Step 3 ：In this case, \(a=1\), \(b=5\), and \(c=9\). We can substitute these values into the quadratic formula to find the roots of the polynomial.

Step 4 ：However, the discriminant of the quadratic equation, \(D = b^2 - 4ac\), is negative in this case, which means the roots of the polynomial are complex numbers, not real numbers.

Step 5 ：Therefore, the roots are of the form \(-b/2a \pm \sqrt{-D}/2a\), where \(-b/2a = -5/2 = -2.5\) and \(\sqrt{-D}/2a = \sqrt{-11}/2 = \pm 1.6583123951777j\).

Step 6 ：Thus, the roots of the polynomial are \(x = -2.5 \pm 1.6583123951777j\).

Step 7 ：Final Answer: The roots of the polynomial are \(x = \frac{-5 - \sqrt{-11}}{2}\) and \(x = \frac{-5 + \sqrt{-11}}{2}\), which correspond to options C and F. Therefore, the correct answers are \(\boxed{\text{C and F}}\).