I'm not 100% sure what the question is asking for (it would have helped to see the whole statement of the problem). But a simple example of a quadratic with complex roots is x\^2 + 1. So an example of a 4th degree polynomial with this as a factor would be x\^4 - 1, which has roots +1, -1, +i, -i, and can be factored as (x-1)(x+1)(x\^2 + 1).