Concept: This question tests the correct spelling of a common English word.
Step 1: Analyze the options for common spelling patterns and sounds
The word in question is "arrogant," meaning having or revealing an exaggerated sense of one's own importance or abilities.
Step 2: Evaluate each spelling
(1) arrogent: Incorrect. The 'a' vowel sound in the last syllable is typically represented by 'a', not 'e', in this word.
(2) arogant: Incorrect. The double 'r' is missing.
(3) arrogant: Correct. This spelling correctly represents the double 'r' and the 'a' in the final syllable.
(4) arogunt: Incorrect. The vowel sound and the ending are incorrect.
The correct spelling is arrogant.