The artery that should be compressed for head and facial hemorrhage is the common carotid artery or facial artery, which is located on both sides of the midline trachea, at the level of the larynx (specifically at the level of the cricoid cartilage) and at the midpoint in front of the sternocleidomastoid muscle. This location is where the common carotid artery bifurcates into the external and internal carotid arteries. By pressing here, the blood supply to the head and facial arteries can be reduced and hemostasis can be achieved. The facial artery is 2 cm forward along the mandible, where the pulsating artery can be felt, and is where the facial artery crosses the mandible.