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The correct answer is A. Common carotid artery. This artery supplies blood to the external carotid artery, which in turn supplies the face, mouth, and nose. It is important to know the pathway of blood flow in the head and neck region to understand the functions of different arteries.
Option B, External carotid artery, is incorrect as it branches off from the common carotid artery and is not itself known as the external maxillary artery.
Option C, Facial artery, is incorrect as it is a separate artery that supplies blood to the muscles and skin of the face, not specifically to the lower region of the face, mouth, and nose.
Option D, Maxillary artery, is incorrect as it is a branch of the external carotid artery, not the artery directly known as the external maxillary artery.