Final answer:
Kublai Khan completed the conquest of China and started the Yuan dynasty after defeating the Song dynasty in 1279.
Step-by-step explanation:
The individual who completed the conquest of China and started the Yuan dynasty was B) Kublai Khan. After fully conquering the Song dynasty in 1279, Kublai Khan established the Yuan dynasty, asserting his claims to supremacy over the entire Mongol Empire. His notable achievement includes re-establishing the Silk Road and promoting cultural exchanges, including hosting the Venetian merchant Marco Polo.