For integers x and y, which of the following MUST be an integer?

You'll have more flexibility in general if you can recognize how to factor in questions like this, but as a fallback, an alternative approach is to plug in some simple numbers for x and y. The right answer needs to always give us an integer result, so if we find that an answer does not give an integer result for some numbers we pick, that cannot be the right answer. And if you plug in x=0 and y=1, you get a negative number under the square root for C and D, so you get something undefined (or 'imaginary' in advanced math language), so those can't be right answers. If you plug in x=1 and y=1, you get √91 for A, and √122 for E, which aren't integers either, so those can't be right. That only leaves D, which does give us an integer in both cases.

If you ever need to use an approach like this, try to focus on the simplest possible sets of numbers, so you don't spent too long doing calculations. Zero (when allowed) can be a very convenient choice, and sometimes, as in this question, just plugging in 1 for everything will get you the answer (we actually could have just plugged in x=1 and y=1 and that would have been enough here - only D gives an integer - but plugging in x=0 rules out two answers so quickly I prefer to do that). At the very least, you usually will rule out three wrong answers by using very simple numbers, and then will only need to plug slightly more complicated numbers into two answer choices instead of five.