A. show that the system of rational expressions is closed under addition, or give an example showing the contrary.

A "rational expression" is a fraction in which the numerator and the denominator are both polynomials. The sum of two such expressions will always produce another such expression, viz., 
p(x)/q(x) + r(x)/s(x) = [p(x)s(x) + q(x)r(x)] / [q(x)*s(x)]. 
The numerator and denominator of the sum will both be polynomials when fully expanded