**Core Concept**
The external carotid artery is a major artery that supplies blood to the face and neck. It has several branches that supply different regions of the head and neck. Understanding the anatomy of the external carotid artery and its branches is crucial for various medical procedures and treatments.

**Why the Correct Answer is Right**
The external carotid artery primarily supplies blood to the face and neck, and its branches include the maxillary artery, superficial temporal artery, and facial artery, among others. The internal carotid artery, on the other hand, is primarily responsible for supplying blood to the brain and the eye. It does not have any branches that supply the face or neck.

**Why Each Wrong Option is Incorrect**
**Option A:** The maxillary artery is a branch of the external carotid artery, supplying blood to the face and the nasal cavity.
**Option B:** The superficial temporal artery is also a branch of the external carotid artery, supplying blood to the scalp and the face.
**Option C:** The facial artery is another branch of the external carotid artery, supplying blood to the face, including the lips, nose, and chin.

**Clinical Pearl / High-Yield Fact**
Remember the mnemonic "MAX" to recall the three main branches of the external carotid artery: Maxillary, Superficial Temporal, and Facial.

**Correct Answer: D. Internal carotid artery**