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The underlined word or phrase (which unfortunately I can't see since it's not indicated in the text) refers to "Jim run for student body president." This phrase is a noun phrase acting as the subject of the infinitive "to suggest." It is not an adverb phrase, as it does not modify a verb, adjective, or another adverb. It is not acting as an adjective since it is not modifying a noun. It is not an appositive phrase because it is not renaming another noun.
Instead, the phrase is best described as a gerund phrase, as it features the verb "run" used as a noun, with "Jim" as its subject. In this sentence, the gerund phrase is the object of the suggestion, indicating the action that was proposed. Therefore, the correct description is:
gerund phrase