We identify a large sample of radio quasars, including those with complex radio morphology, from the Sloan Digital Sky Survey and the Faint Images of the Radio Sky at Twenty cm ( FIRST). Using this sample, we inspect previous radio quasar samples for selection effects resulting from complex radio morphologies and adopting a positional coincidence between radio and optical sources alone. We find that 13.0% and 8.1% of the radio quasars do not show a radio core within 1.2 '' and 2 '', respectively, of their optical position and thus are missed in such samples. Radio flux is underestimated by a factor of more than 2 for an additional 8.7% of the radio quasars. These missing radio-extended quasars are more radio-loud, with a typical radio-to-optical flux ratio, namely, radio loudness RL greater than or similar to 100 and radio power P greater than or similar to 10(25) W Hz(-1). They account for more than one-third of all quasars with RL > 100. The color of radio-extended quasars tends to be bluer than that of radio-compact quasars. This suggests that radio-extended quasars are more radio-powerful sources, e. g., Fanaroff-Riley type 2 sources, rather than the compact sources viewed at larger inclination angles. By comparison with the radio data from the NRAO VLA Sky Survey, we find that for sources with total radio flux less than 3 mJy, low surface brightness components tend to be underestimated by FIRST, indicating that lobes in these faint radio sources are still missed.