Final answer:
The phrase 'They danced and said goodnight' is a complete sentence, not a run-on sentence, as it includes a subject and two verb phrases correctly connected by the conjunction 'and.'
Step-by-step explanation:
The question asks whether the phrase "They danced and said goodnight." is a complete sentence or a run-on sentence. A run-on sentence occurs when two or more complete sentences are joined without proper punctuation or conjunctions. Examining the phrase, it contains a subject and a verb together with a coordinating conjunction 'and,' which connects the two verb phrases 'they danced' and 'said goodnight.' Therefore, it is not a run-on sentence, as it correctly uses the conjunction 'and' to join these clauses. It is a complete sentence that communicates a clear idea with proper structure.