My solution for Vout is the same as yours. However, I do not understand your method. My method is to realize that the voltage difference between the op-amp inputs must be zero.
At first I did not understand why you calculated Re since it is not relevant to the solution. But now I can see you don't use Re anywhere in your solution.
Then I did not understand the marked paths labeled with circled '1' and '2' going from Vin1 to GND and from Vout to GND. That is still a mystery.