Which of the following is the complete list of roots for the polynomial function f(x) = (x2 + 2x - 15)(x2 + 8x + 17)?A) -5,3B) -5,3,-4,+I,-4,-iC) -5,3,-4,+I,4+ID) -4+i , -4 -i

The first one factors easily.  The first quadratic, I mean.  The 2 numbers that add up to +2 and at the same time multiply to -15 and 5 and -3.  So those are 2 of the 4 roots we have.  The second quadratic does not factor so nicely.  You need to put that into the quadratic formula to solve.  [tex]x= \frac{-8+/- \sqrt{8^2-4(1)(17)} }{2} [/tex]  which simplifies to  [tex]x= \frac{-8+/- \sqrt{64-68} }{2} [/tex].  That gives us a negative radicand and that's a problem.  [tex]x= \frac{-8+/- \sqrt{-4} }{2} [/tex].  Since -1 is equal to i^2, we can rewrite to begin dealing with the negative properly.  [tex]x= \frac{-8+/- \sqrt{-1(4)} }{2} [/tex].  Replacing -1 with i^2 gives us  [tex]x= \frac{-8+/- \sqrt{i^2(4)} }{2} [/tex].  i^2 has a perfect root of i in it, and 4 has a perfect square of 2 in it, so we simplify more to  [tex]x= \frac{-8+/-2i}{2} [/tex].  The 2 in the denominator reduces with the numerator to give us a final 2 roots that are  x = -4 + i,  and x = -4 - i.  Taking all those roots together, we find that the solution to our problem is choice B (although I believe you put some extra commas in there on accident).