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

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

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

Answer

Explanation:

Step1: Solve $x^{2}+6x + 8=0$

Factor the quadratic equation: $(x + 2)(x + 4)=0$. Then, by the zero - product property, $x+2 = 0$ or $x + 4=0$. Solving these gives $x=-2$ and $x=-4$.

Step2: Solve $x^{2}+6x + 13=0$

Use the quadratic formula $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$ for the quadratic equation $ax^{2}+bx + c = 0$. Here, $a = 1$, $b = 6$, and $c = 13$. First, calculate the discriminant $\Delta=b^{2}-4ac=(6)^{2}-4\times1\times13=36 - 52=-16$. Then $x=\frac{-6\pm\sqrt{-16}}{2}=\frac{-6\pm4i}{2}=-3\pm2i$.

Answer:

C. $-2,-4,-3 + 2i,-3 - 2i$