Final answer:
Spain's two great viceroyalties in the New World during the 1530s and 40s were the 1. Viceroyalty of New Spain and the Viceroyalty of Peru, with their capitals at Mexico City and Lima respectively.
Step-by-step explanation:
In the 1530s and 1540s, the Spanish government divided its New World territories into two major areas known as viceroyalties. The question pertains to the identification of what were Spain's two great viceroyalties in the New World during that period. The correct answer to this question is: 1) Viceroyalty of New Spain and Viceroyalty of Peru.
The Viceroyalty of New Spain included areas such as Mexico, Central America, and parts of what is now the United States, with its capital at Mexico City. The Viceroyalty of Peru consisted of territories like modern Peru, Chile, Bolivia, and Ecuador with its capital in Lima.
These viceroyalties were overseen by viceroys, who held significant authority and were assisted by councils such as the Audiencia. Unlike English colonies, there were no elected officials or legislatures since the Spanish Crown appointed all administrative positions.