Final answer:
The ophthalmic division of the trigeminal nerve innervates the lateral walls of the nasal cavity and its mucous membrane.
Step-by-step explanation:
The nerve that innervates the lateral walls of the nasal cavity and its mucous membrane is the ophthalmic division of the trigeminal nerve (option A).
The trigeminal nerve is the fifth cranial nerve and is responsible for providing sensory innervation to the face, including the skin, muscles of mastication, and the nasal cavity. The ophthalmic division of the trigeminal nerve specifically innervates the forehead, upper eyelid, and lateral walls of the nasal cavity.
The other options listed - abducens nerve, trochlear nerve, and vestibulocochlear nerve - are not involved in innervating the lateral walls of the nasal cavity.