2006 AMC 10B Problem 14

Let a and b be the roots of the equation x^2-mx+2=0. Suppose that a+\frac{1}{b} and b+\frac{1}{a} are the roots of the equation x^2-px+q=0. What is q?

\mathrm{(A) \ } \frac{5}{2}\qquad \mathrm{(B) \ } \frac{7}{2}\qquad \mathrm{(C) \ } 4\qquad \mathrm{(D) \ } \frac{9}{2}\qquad \mathrm{(E) \ } 8

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.