Here are my thoughts. I want to show that every submodule of this tensor product is finitely generated (I think that is easier than trying to show it satisfies the ascending chain condition) Also, because $A$ is a noetherian ring, $A[x]$ is also noetherian by Hilbert basis theorem. I want to take a submodule of $M\otimes A[x]$ and show that it projects to a submodule of $M$ and a submodule of $A[x]$ Then use the generators of each of these submodules, tensor them together to show that I now have generators for my original submodule again. Is this the right idea?
Also, how does the action of $A[x]$ work on $M\bigotimes_A A[x]$? How is it an $A[x]$ module? Would it be
$a(x)*(m\otimes p(x)) = m\otimes p(x)a(x)?$
Source: Old qual exam Spring 1992 5a.