factoring with complex numbers

Another question...well more of a clarification of a question. Here is the question:

Find the values of the real numbers *p* and *q* if *x^2+1* is a factor of the polynomial P*(x)=x^4+px^3+2x+q**.* Hence, factorise over **R** and over **C.**

I've determined that:

*x=i*

therefore after simplifying:

*P(i)=(1+q)+i(2-p)*

But now a little bit unsure of what to do. Does this mean that q=-1, and p=2

Then how do I factorize over **R** and **C?**

Thanks

mmm, think I just figured it out...Im gonna use synthetic division to find the remaining roots (assuming it works out nicely) but still not sure what the deal is with **C**