The answer to your question would be that the underlined phrase that is an adjective prepositional phrase is the following one: Robert Fulton's invention of the steambot gives him a prominent place in the history of transportation.
Prepositional phrases can function in three ways: they can function as a noun, an adjective, or as an adverb. When the prepositional phrase functions as an adjective, it gives us additional information about a noun or pronoun in the sentence. It tells us which one, what kind, or how many.
If the object of the preposition is related to another noun or pronoun, then the prepositional phrase is functioning as an adjective. In this case, "the steambot" (object of the preposition) is related to the "invention" (another noun), it is the invention by Robert Fulton.
