This one certainly isn't immediately obvious. The first thing to note is that your x^2 + 9/x^2 sure looks like a quadratic, and in fact it looks quite a bit like (x - 3/x)^2 squared. Let's take these suppositions and get to work.

First, let's just square (x - 3/x) to see what happens: by binomial expansion, we see that (x - 3/x)^2 = x^2 - 6 + 9/x^2.

Now this is really close. If we simply add six to it, we get the original x^2 + 9/x^2 statement--this is the hard part to notice! Let's see what happens:

x^2 - 6 + 9/x^2 + 6 = 31

Now let's substitute in the bracketed term to see what happens--this is another way to write exactly the same thing:

(x - 3/x)^2 + 6 = 31

From here, it becomes clear that we can just move the six to the right side:

(x - 3/x)^2 = 25

And from here, we can simply take the root of each side. While the technical \sqrt{25} will be +5 or -5, note that we don't have the -5 option.

The answer is D.