Final answer:
Rome integrated Samnium by forming alliances, offering citizenship and military service.
Step-by-step explanation:
The allied integration of Samnium by Rome was achieved through various means. Rome formed alliances with the Samnites and offered them citizenship and military service, integrating them into the Roman Republic. This allowed Rome to potentially expand its military might with each victory. Additionally, Rome established colonies in recently conquered Samnite lands, built roads to link them to Rome, and transformed former enemies into loyal allies who received Roman citizenship.