Final answer:
The state that borders Vermont with the Connecticut River on its boundary and contains the Merrimack, Androscoggin, Piscataqua, and Salmon Falls rivers is New Hampshire.
Step-by-step explanation:
The state that has the Connecticut River on its border with Vermont, as well as the Merrimack River, Androscoggin River, Piscataqua River, and Salmon Falls River is New Hampshire.
These rivers are significant geographical features in the region. The Connecticut River serves as the boundary between New Hampshire and Vermont, while the other rivers mentioned are located within New Hampshire but can also mark boundaries or provide water resources to other nearby states.