Read the sentence. the hotel guests enter through a beautiful courtyard. which is the prepositional phrase? the hotel guests enter a) enter through b) through a beautiful courtyard c) through a beautiful

The correct answer is:  [B]:  "through a beautiful courtyard" .

Note:  A 'prepositional phrase' must start with a preposition; and in addition to including the preposition; the 'prepositional phrase must include its object as well.
   
So, "Choice [A]:  "enter through" ; is clearly incorrect, since it does not start with a preposition.   Furthermore, there is no object.

Consider:  "Choice: [C]:  "through a beautiful" .  This choice is incorrect; since there is no object; so this choice is incorrect because it is "incomplete" for the purpose of meeting the criteria for a "prepositional phrase.

Choice: [B]:  "through a beautiful courtyard" ; is correct;  since this is modifying phrase that: 1) begins with a preposition—"through" ; and includes its object—"courtyard"—and as such, meets the criteria for a "prepositional phrase".