ROC of the function in the problem 9.14 of Oppenheim's Signals and Systems textbook I have solved the problem 9.14 in Oppenheim's Signals and Systems textbook, but my solution and the one in Slader is different. Problem is given above. And Slader solution is here.

I have also attached my solution below. Laplace transforms are the same but ROC in the Slader solution and mine is different. My question is how can this Laplace transform have an ROC as Re{s}>0. This should be wrong since this region contains the two poles of the Laplace transform which should not be the case for a rational Laplace transform.

Thanks in advance.
You're right, the ROC of the Laplace transform of a two-sided signal is a strip in the complex plane. In your case, the imaginary axis is inside the ROC, and the ROC is limited by the poles in the right and left half-planes. If the ROC were the right half-plane, the signal would be right-sided, which is clearly not the case.