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%.
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