  Let $S$ be a parametric surface in $\proj{3}$ given as the image of $\phi:
\proj{1} \times \proj{1} \to \proj{3}$. This paper will show that the use of
syzygies in the form of a combination of moving planes and moving quadrics
provides a valid method for finding the implicit equation of $S$ when certain
base points are present. This work extends the algorithm provided by Cox for
when $\phi$ has no base points, and it is an analogous to some of the results
of Bus\'{e}, Cox and D'Andrea for the case when $\phi: \proj{2} \to \proj{3}$
has base points.
