They're gonna go to budget reconciliation, which I believe would set a very bad precedent, because essentially -- if they could do it, and I don't know if they can do it, but if they could do it -- what you have done, effectively, is to take away the filibuster in the United States Senate. . . . So, therefore, you have 51 votes in the House and 51 votes in the Senate. That is not what our Founding Fathers had in mind. That is not the constitutional process (emphasis added).Reality
Article I of the Constitution establishes and delegates power to Congress. Article I is silent regarding how many votes a particular bill must receive before it is sent to the President and with respect to the procedural rules for each chamber. Instead, the Constitution generally allows each chamber to determine its own rules (with some very specific exceptions).
Nevertheless, Article I clearly describes the actions of Congress that require a supermajority. For example, overriding a veto requires a 2/3 vote of each house. Also, a decision to convict the President during the impeachment process requires a 2/3 vote of the Senate. The Constitution, however, does not create the filibuster, even though it has a long history. The filibuster is a Senate rule -- not a constitutional mandate.
Finally, it is unclear where Quayle gets the "51 votes in the House" figure. A simple majority of the House consists of at least 218 votes.