  The process pp -> top anti-top bottom anti-bottom + X represents a very
important background reaction to searches at the LHC, in particular to top
anti-top H production where the Higgs boson decays into a bottom anti-bottom
pair. A successful analysis of top anti-top H at the LHC requires the knowledge
of direct top anti-top bottom anti-bottom production at next-to-leading order
in QCD. We take the first step in this direction upon calculating the
next-to-leading-order QCD corrections to the subprocess initiated by quark
anti-quark annihilation. We devote an appendix to the general issue of rational
terms resulting from ultraviolet or infrared (soft or collinear) singularities
within dimensional regularization. There we show that, for arbitrary processes,
in the Feynman gauge, rational terms of infrared origin cancel in truncated
one-loop diagrams and result only from trivial self-energy corrections.
