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1.) Would having a Republican-Democratic presidency benefit the effectiveness of our government or not?
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3.) What kind of foreign/domestic policy would result from a Republican-Domestic office? Or how would this affect existing policies (such as NCLB or the Patriot Act) ?
I'll answer questions 1 & 3 together since #3 is a specific version of #1. I don't believe that you would see any diluting of a specific bill such as NCLB or the PATRIOT Act, should you have a Republican-Democratic ticket. I believe that you are hinting that NCLB and the PATRIOT Act, which are two key Republican provisions would be challenged by the other party? It turns out that both parties have been of one mind on these two. When NCLB was passed in 2001,
six republicans voted no while only two democrats opposed the measure. Those Republicans by the way were Voinovich, Inhofe, Helms,Kyle, Nickels, and Bennett. The bill passed by a wide margin, 91-8. The same kind of unanimous support was found with the PATRIOT Act of 201. How did the democrats do?
46 voted for it, one opposed, and one didn't cast a vote. In light of these numbers, it can be seen that the effect of having a democrat on board who can support the line for the most part, is feasible. Now I'm sure that the recent votes on these bills has changed somewhat. At the same time, you have an interesting coalition in both parties that oppose them. That is why you have the likes of Kucinich and Bob Barr opposing the PATRIOT Act.

Or why GOP legislators in Virginia oppose NCLB and the lone person to vote against a rebellion in Virginia on NCLB was a democrat.