  Combining radio observations with optical and infrared color selection --
demonstrated in our pilot study to be an efficient selection algorithm for
finding red quasars -- we have obtained optical and infrared spectroscopy for
120 objects in a complete sample of 156 candidates from a sky area of 2716
square degrees. Consistent with our initial results, we find our selection
criteria -- J-K>1.7, R-K>4.0 -- yield a ~50% success rate for discovering
quasars substantially redder than those found in optical surveys. Comparison
with UVX- and optical color-selected samples shows that >~ 10% of the quasars
are missed in a magnitude-limited survey. Simultaneous two-frequency radio
observations for part of the sample indicate that a synchrotron continuum
component is ruled out as a significant contributor to reddening the quasars'
spectra. We go on to estimate extinctions for our objects assuming their red
colors are caused by dust. Continuum fits and Balmer decrements suggest E(B-V)
values ranging from near zero to 2.5 magnitudes. Correcting the K-band
magnitudes for these extinctions, we find that for K <= 14.0, red quasars make
up between 25% and 60% of the underlying quasar population; owing to the
incompleteness of the 2MASS survey at fainter K-band magnitudes, we can only
set a lower limit to the radio-detected red quasar population of >20-30%.
