Then checks to see 'how closely' this quadratic factor divides into the polynomial. This is done repeatedly until 'how closely' is 'close enough'. The quadratic factor that fits now has two of the roots of the polynomial. The pair of roots may real, or complex.
Procedure is repeated to look for the next quadratic factor.
Should have mentioned that if the degree of the original is ODD, then there will be at least one real root. This one real root is found by successive approximation. After that, you can solve for the roots of polynomials all day long.