The student knew that +2i was another solution, but didn't know how to use the two solutions to answer the question. Although the student knew how to do polynomial long division from a previous lesson, we first looked at a graph of the function to see if this function had 4 complex zeros or 2 complex zeros and 2 real zeros. From the graph, we can see that there are real zeros at -3 and 5.
Next, we divided (x - 2i)(x + 2i) = x2 + 4 into P(x) and found our quotient to be x2 - 2x - 15. This quadratic factors into (x + 3)(x - 5), which agrees with what we saw in the graph.
"When am I ever going to use this?" the student asked me. I told him truthfully that he would probably never personally use this outside of another math class. Then he asked a different question, "Where are complex numbers used?" I told him in engineering - working with electronics and with fractals. Then I referenced the Genesis planet in Star Trek and computer graphics as a use for fractals. (After posting this blog, I'll be sending the student the video link.)
Special Note: Much of the math I teach, other than statistics, will never be used by many of my students. So, why teach it? Of course I want students to learn math concepts, but more importantly I want them to notice, wonder and ask questions. Why does this polynomial go up on the left and the right and the other one goes down on the left and up on the right? What about the numbers in the polynomial make them do that? Helping students make connections between ideas and noticing relationships. Having them make and test conjectures. Having them say "What if?" or "Does this always happen?" This type of thinking is what I want my students to do beyond my classroom. The mathematics will be important to some of them later. But for all my students, I hope the math we discover together encourages them to be curious and to think deeply.