In my most recent essay, I had used the term "fleshed out" but my teacher had circled it. I called her over to ask why it was circled and she said that "Flushed out is the correct term to use." I googled it up in front of her and it had said that I was correct and she was wrong. She then said that either way it was informal and is not to be used in a formal essay. I ask this question because I could swear that it was formal, but now that she is saying it isn't I'm starting to question myself. Thanks :)
Formality is a quality determined by context and audience. It is an intangible measure that is hard to pin down, since individual readers may disagree about what they expect. So my answer won't be to say it's formal or informal, but to describe what someone might consider in evaluating how formal it is.
- Etymology and formality. There are lots of examples of synonyms in English coming from multiple languages of origin. For instance, groups of synonyms like regal/royal/kingly, interrogate/question/ask, and ascend/mount/rise all follow a pattern: Latin-origin/French-origin/Old English-origin. Their origins tend to determine how they're used: the Latin and French words tend to be more formal in tone. This tendency (not a rule; you can probably find exceptions) carries over to other languages. For flesh out I can think of alternatives: elaborate (Latin), explicate (Latin), analyze (Greek via French and Latin), and expound (French). So it may be that your teacher would think of most of those other terms as formal.
- Jargon and formality. You know those words I just listed? They constitute a kind of academic jargon. These two sources (both PDFs) provide some guidance on what words are used in academic contexts. They are fundamentally descriptive and non-comprehensive as documents. The first one models a dislike of "casual turns of phrase" when another expression is available: no to "chalk that up" or "not-terribly-difficult." The second is a lexicon about writing, but similar principles prevail: analysis and interpretation are used rather than more colloquial possibilities. It may be that your teacher prefers established academic verbs to what she sees as a casual turn of phrase.
- Recency. Flesh out, in its current usage, is a new verb phrase. The Oxford English Dictionary only attests flesh with out from the late 19th century. (Prior uses include flesh over and flesh with.) The OED marks this usage as metaphorical; Wiktionary marks it as "idiomatic," signalling that the peculiarity of the usage hasn't been lost. In this Ngram graph, note that "flesh out" becomes more common after 1960 and eclipses "flesh with" around 1980, suggesting that it may have been in wide use in academic contexts for a relatively brief time. Your teacher may therefore not be aware that academics do use the term.
In short, I would only blink at the phrase if it were used in a non-idiomatic way, but there are a few reasons why someone might read it as informal. As with any writing, consider your audience and think about the risks of either option.
I think the previous answers are excellent but may have missed at least part of the teacher's point. "Fleshed out" and "flushed out" have entirely different meanings - both are correct and perfectly acceptable in the most formal of contexts.
"Flesh out" derives from the idea of adding flesh to the bones (for example, in a life drawing). Figuratively it means to add substance to an argument or description, to take an exposition further, with more depth and scope.
"Flush out" derives from hunting, where, for example, a dog runs into the undergrowth and rouses birds so that they fly up into the air and can be shot. This use of the word flush is probably the oldest etymologically, with other uses (water, reddening of the face, etc.), deriving from it. Figuratively, it can be used to mean teasing or forcing out evidence, truths, and so on from a text, identifying the underlying meanings in a book or a movie.
So, I think this is more about specific meaning than formality. So, the teacher may have been right. However, I would strongly disagree with her about formality, if that really was her main point. Flush out is an ancient expression and flesh out, as carefully explained in the previous answer, is very well established; they are in no way synonyms as they have entirely different meanings.
More generally, great writers embrace the use of colloquial and idiomatic metaphor, where the meaning is thus best expressed and they add colour - that is not the same as using slang, contractions or vulgarisms.
After all, a very large proportion of even our most formal language is, at root, figurative. So, in this last sentence: "after" from Old Norse for back, "proportion" from the Latin for a portion (of food etc), "large" from physical size, "language" from the Latin for tongue, "root" from the part of a plant, "figurative", from the Latin for "to shape".