Lets say we have a polynomial with only complex roots. an example would be $$9x^6-4x^5+x^4-7x^3+8x^2-8x+7$$ How would one go about finding the roots to this polynomial and factoring it. I know this may seem like just asking for an answer, but I would really like to know the process behind it because I couldn't find anything online that would seem to work.
For the case where the degree is 3 or 4, there are formulae giving the roots. However, these formulae cause some nasty numerical problems, so they have to be implemented very carefully, as discussed here. Actually, even the quadratic formula has numerical problems if you don't code it correctly.
For degrees higher than 4, there are no general closed-form formulae, so you have to use numerical methods. There is a survey of available methods here. You shouldn't have to write any code yourself -- the work has already been done by experts, and software libraries are widely available.