  We present the first calculation of $Z+4$ jet production with heavy quark
flavor identification at the Tevatron $p\bar p$ collider. The $Z + 4$ jet
channel is especially interesting as a normalizer for the $W +4$ jet background
to top quark signals, as a background to a possible $t\to cZ$ flavor-changing
neutral-current (FCNC) decay signal, and as a background to missing-$p_T$
signals from gluino pairs. We also calculate the contributions to $W+4$ jet
production from all the different heavy-flavor final states. The MADGRAPH
program is used to generate all leading order subprocess helicity amplitudes.
We present Monte Carlo results with separation and acceptance criteria suitable
for the Tevatron experimental analyses. The dependence of the cross sections on
experimental cuts and the theoretical ambiguities due to the scale dependence
are discussed. The predicted ($W+4$ jet)/($Z+4$ jet) ratio is insensitive to
most of these choices.
