What should be my first. Initial thought when I look at this problem? What should I be looking for?

Mar, the first thought should be: I should have remembered my algebraic identities from high school. For example, (x+y)^2 = x^2 + y^2 + 2xy. In the problem, you can rearrange the given expressions as x^2 + y^2 + 2xy in order to get (x+y)^2, so that it would be easy to solve for x+y, which is the question. Since we are told that x^2 + y^2 = 2ab and 2xy = a^2 + b^2, then (x+y)^2 = 2ab + a^2 + b^2. However, using the same algebraic identity, we recognize that the right hand side is nothing but (a + b)^2.

Then we take the square roots of both sides, and get x+y = a+b.