The correct spelling of the phrase meaning to study at a questionable or fraudulent institution is "study at quack." The word "quack" is spelled with a "qu" digraph, representing the sound /kw/. The vowel sound in "quack" is a short "a" /æ/, followed by a consonant sound /k/. When combining the phrase with "study at," it forms a closed syllable with a long "u" sound /u/. It is important to use the correct spelling to avoid confusion and ensure accurate communication.