If the entire body of the question was just:
I just entered "\frac{x^2}{x^4+1}" into the search box, with unfruitful results. Can things like this be done?
Then it is frightfully short. That's the main obstacle here, is that a question that is barely even two sentences usually is an indicator of a barebones and poor quality question. We have a "quality standards" filter meant to block things which are likely to be incomplete or incomprehensible to the answerers - too short to have significant data, filled with spelling errors or lacking punctuation, gratuitous use of inane abbreviations, is just a spam link... that sort of thing. It is basically meant to prevent the low-tier junk from ever showing up so that the community doesn't have to waste time cleaning it up.
That isn't to say yours is a bad question. You're simply being direct and the whole Meta question is mostly there. However, it still can see some improvement.
For starters, do you know that the specific exponential fraction you're searching for exists on the site? If so, can you hypothesize that the lack of results comes from searching for something inside of a MathJax statement rather than a complete MathJax statement?
It's minor extra data, but it is data that helps answerers address your concerns faster. Meta questions can always use extra info, just like main Q&A questions. Saving the answerers the need to test things out (by, say, linking an example question that does have said exponential fraction in it) will go lengths to help figure things out if people don't happen to know the answer offhand.
I strongly recommend that if you run into the quality standards filter, that you just check to see if there's any, any extra data you can add that can help answerers. To take an example, the question body here is ridiculously shallow (and I don't know how it got past the quality filter, excepting possibly some kind of empty set abuse that was removed within the 5 minute grace period), and then there is a bunch of clarification in the comments. Including all that clarification in the initial question body would've been far more productive and would've avoided the comment conflict on duplication. The more data available from the start, the less people have to ask from the start.
To save you the trouble, by the way, I believe this Meta discussion points out that our LaTeX searching capabilities are quite sub-par, so unfortunately it appears that it is not currently possible to search for an expression like you were attempting. I do not know whether or not Kenny's answer provides a sufficient tool or not, but hopefully it still works. ♪
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? ;) $\endgroup$