Hmmmm.....nothing obviously superior that I'm aware of. You could use the quadratic equation. The advantage of that is that it eliminates guesswork. The disadvantage is it's tedious and you have to know the square root of 961.
How did you end up getting 15*16=240? Guessing and checking? If so, there is a way to improve on that, but first some background info to make sure we're on the same page. This is going to be kind of long, but once you see the trick you'll see it's a lot faster than guessing and checking.
If we list the factors of some perfect square, let's say 36, in pairs:
1,36
2,18
3,12
4,9
6,6
Note that in each pair, one is always less than the square root of 36 and one is always greater until we get to 6,6 where they are both exactly equal to the square root of 36. As we move from the top of the list to the bottom, the number on the left is approaching the square root of 36 from the bottom and the number on the right is approaching the square root of 36 from the top. But except for 6 and 6, there is always one number greater than 6 and one number less than 6 in each pair.
Now let's look at an integer that is not a perfect square, 72:
1, 72
2, 36
3, 24
4, 18
6, 12
8, 9
For a reference point, the square root of 72 is approximately 8.49. Note that the same thing is happening: one factor is always less than 8.49 and one is always greater than 8.49. Now, 72 has two factors that differ by 1: 8*9. Note that this is the ONLY possible pair that could have worked. The square root is about 8.49, so if 72 were going to have two factors that differ by 1, it would HAVE to be 8 and 9 to keep with the rule of each pair of factors having one number greater than the square root and one number less than the square root.
Let's say we want to find out if some other number has factors that differ by 1. If the number were 87, we could reason as follows: 9^2=81 and 10^2=100. Therefore the square root of 87 must be between 9 and 10. That is, 9 point something. So, if 87 DOES have factors that differ by 1 they must be 9 and 10. However, 9*10=90, not 87, so that must mean 87 does NOT have two factors that differ by 1.
So if we had to factor n^2-n-87, we would know for sure that it is impossible, because we've just proven that 87 doesn't have a pair of factors that differ by 1.
Now, in the case of 240, 15^2=225 and 16^2=256, so the square root of 240 must be between 15 and 16. So if 240 CAN be factored as two integers that differ by 1, it must be 15 and 16. Then you just have to verify that they work by multiplying them together.
So, to summarize, if an integer, n, does have factors that multiply to n and have a difference of 1, one of them must be first integer above the square root of n and one must be the first integer below the square root of n.
