  We predict the differential cross sections for production of the X(3872) at
the Tevatron and the Large Hadron Collider from both prompt QCD mechanisms and
from decays of b hadrons. The prompt cross section is calculated using the
NRQCD factorization formula. Simplifying assumptions are used to reduce the
nonperturbative parameters to a single NRQCD matrix element that is determined
from an estimate of the prompt cross section at the Tevatron. For X(3872) with
transverse momenta greater than about 4 GeV, the predicted cross section is
insensitive to the simplifying assumptions. We also discuss critically a recent
analysis that concluded that the prompt production rate at the Tevatron is too
large by orders of magnitude for the X(3872) to be a weakly-bound charm-meson
molecule. We point out that if charm-meson rescattering is properly taken into
account, the upper bound is increased by orders of magnitude and is compatible
with the observed production rate at the Tevatron.
