What are two methods by which amendments to the U.S. Constitution can be proposed?

Amendments to the U.S. Constitution can be proposed through two established methods: a two-thirds vote of both the House of Representatives and the Senate, or by a national convention called by two-thirds of the state legislatures. The first method emphasizes the congressional authority in the legislative process, ensuring that a significant majority consensus is achieved within Congress before navigating the long and rigorous amendment process.

The second method, through a convention, reflects the framers’ intention that the states should also have a voice in the constitutional amendment process, allowing for significant changes when a large majority of states express the need for an amendment. This method has never been used, but it exists as a potential avenue for proposing amendments, demonstrating the flexibility and adaptability of the Constitution.

The other choices do not align with the constitutional requirements for proposing amendments. Popular votes, judicial rulings, executive action, state ballot initiatives, and congressional hearings are not valid methods for constitutional amendment proposals. Thus, the correct answer emphasizes the established and constitutional pathways through which amendments can be initiated.