Final answer:
The Spanish did not claim the St. Lawrence River, which was claimed by France, nor did they follow the Mississippi from its source to its mouth. They did, however, gain great wealth and claimed lands in the southeastern United States.
Step-by-step explanation:
Out of the statements given about the Spanish exploration and conquests in the New World, two are NOT true:
The Spanish explorers claimed the St. Lawrence River.
The Spanish explorers followed the Mississippi from its source to its mouth.
These statements are incorrect because it was the French explorers, not the Spanish, who claimed the St. Lawrence River, with Jacques Cartier naming it New France. Furthermore, it was not the Spanish but other explorers who navigated the full length of the Mississippi. The Spanish did indeed gain great wealth from their conquests in the New World, particularly from gold and silver mining. Moreover, Spanish explorers such as Hernando de Soto did claim large stretches of land in what is now the southeastern United States.