I am attempting to factor a relatively benign looking polynomial of degree 4. I have tried to use synthetic division to factor this. I was hoping to be able to get a remainder of zero at some point. I don't quite know how this works but I suspect that if perhaps I extend things to complex numbers I might have some luck. For this same reason, I am suspecting that all of the roots are may be complex. Can someone tell me how one deals with this?
In general, something like synthetic division along with the Rational Root Theorem might not be the worst idea, assuming there is a real root. Otherwise, it might be a product of two irreducible quadratic polynomials in which case you may want to look at something like factor-by-grouping. In the worst scenario, you have an 'exotic' polynomial and will have to solve for the roots. Luckily, mathematicians in the 1500s managed to find formulas - albeit of the headache inducing variety - to do exactly this, see the Wiki page for quartic functions.
Using a CAS can greatly reduce the headache.