1. <em>(After several minutes,) </em><em>I finally located my car keys. </em>The phrase in parenthesis is a <u>"prepositional phrase" (option A)</u>. This prepositional phrase is formed by the preposition "after" plus the adjective "several" and the noun "minutes".
2.<em> Mikayla's homeroom teacher, </em><em>(Mr. Jones.) </em><em>is proud of her</em>. The phrase in parenthesis is an <u>"appositive" (option C)</u>. The appositive "Mr. Jones" provides additional information about the noun "teacher".
3. <em>Henry's desire </em><em>(to win)</em><em> is obvious</em>. The phrase in parenthesis is an <u>"infinitive" (option B)</u>. The infinitive phrase in this case is used as an adjective which modifies the word "desire".
4. <em>(Talking)</em><em> is not allowed in class.</em> The phrase in parenthesis is a <u>"gerund" (option B)</u>. A gerund phrase includes a word ending in "ing" and functions as a noun.
5. <em>Josie's project was a failure,</em> <em>having not prepared properly.</em> This sentence contains a <u>"dangling modifier" (option B)</u>. A dangling modifier is a a type of misplaced modifier, but it does not have anything to modify in the sentence; that's why the sentence should be restated so as to include the word/s that the modifier intends to describe.