Final answer:
Checking for symmetry of facial movement helps detect injury to the facial nerve after a carotid endarterectomy.
Step-by-step explanation:
Checking for symmetry of facial movement by asking a patient to smile is important after a carotid endarterectomy because it helps to detect injury to the facial nerve. The facial nerve controls the muscles of facial expression, and any compromise in its function can result in asymmetrical movement or facial paralysis. By observing the patient's ability to smile, healthcare professionals can assess the integrity of the facial nerve and determine if any injury or damage has occurred during the surgery.