hmmmm ok so lets see if i got this right...

In order for (X^2 + 2X + 5)/(x - 3) to be an interger, that means when you do the long division, there is no remainder. Now if you actually try doing the long division of polynomials (here I am assuming you know how to divide polynomials) you are going to notice that there is a remainder of 20/(x-3) now in order for it to be an integer answer, the remainder has to be an integer. The highest number that x can be that would make the remainder an integer would be 23. I think this should work...