  In a survey of quasar candidates selected by matching the FIRST and 2MASS
catalogs, we have found two extraordinarily red quasars. FIRST J013435.7-093102
is a 1 Jy source at z=2.216 and has B-K > 10, while FIRST J073820.1+275045 is a
2.5 mJy source at z=1.985 with B-K = 8.4. FIRST J073820.1+275045 has strong
absorption lines of MgII and CIV in the rest frame of the quasar and is highly
polarized in the rest frame ultraviolet, strongly favoring the interpretation
that its red spectral energy distribution is caused by dust reddening local to
the quasar. FIRST J073820.1+275045 is thus one of the few low radio-luminosity,
highly dust-reddened quasars known. The available observational evidence for
FIRST J013435.7-093102 leads us to conclude that it too is reddened by dust. We
show that FIRST J013435.7-093102 is gravitationally lensed, increasing the
number of known lensed, extremely dust-reddened quasars to at least three,
including MG0414-0534 and PKS1830-211. We discuss the implications of whether
these objects are reddened by dust in the host or lensing galaxies. If reddened
by their local environment, then we estimate that between 10 and 20% of the
radio-loud quasar population is reddened by dust in the host galaxy. The
discovery of FIRST J073820.1+275045 and objects now emerging from X-ray surveys
suggests the existence of an analogous radio-quiet red quasar population. Such
objects will be entirely missed by standard radio or optical quasar surveys. If
dust in the lensing galaxies is primarily responsible for the extreme redness
of the lensed quasars, then an untold number of gravitationally lensed quasars
are being overlooked.
