Final answer:
Charlemagne earned the support of the papacy through his close relationship with Pope Leo III and the coronation as Emperor of the Romans. He also promoted Christianity, reformed the Church, and supported education and scholarship.
Step-by-step explanation:
Charlemagne earned the support of the papacy through several actions. One of the key factors was his close relationship with Pope Leo III, who crowned Charlemagne as Emperor of the Romans in 800 AD. This act solidified the alliance between Charlemagne and the papacy, as it gave both political and religious legitimacy to Charlemagne's rule. Additionally, Charlemagne actively promoted the Christian faith, reorganized the Church, and sponsored education and scholarship, which endeared him to the papacy.