The pattern is pretty easy to remember: お + verb stem + します. The only other thing to remember is that you ONLY USE IT FOR VERBS WHERE YOU YOURSELF ARE THE SUBJECT. A few examples: お返しします (I humbly return something to someone), お断りします (I humbly refuse), お持ちします (I humbly carry something), お借りします (I humbly borrow something).
While お願いします is a form of keigo, it has other more important uses than purely just as a humble request. It is, as we say in English about “please,” the “magic word.” It’s almost more important than please in English – it’s please and thank you all wrapped into one.
This is purely theoretical, but I’m willing to bet that people in the States would be more offended by people not saying “thank you” than by people not saying “please.” I’m equally willing to bet that people in Japan are more offended by a lack of お願いします rather than a lack of ありがとう.
願う (ねがう) means to hope or request. I’m confident that it should always be used following a request to someone equal or above you, and it’s worth tacking on to all requests so you don’t end up looking like an asshole. Like a please or a thank you, it softens whatever request you made and shows that you you truly appreciate the effort that they, in this case, will go through. You can add a よろしく on to the front to reemphasize the request (by drawing it out through additional syllables, which always means “more polite” in Japanese). In the case of people on an equal level you can opt for よろしく on it’s own; I have a feeling that the Japanese teacher of English I worked with used this with his students as a sort of joke when he handed them assignments. (よろしくね *cruel laughter*)
In conclusion, よろしくお願いします is often grossly misunderstood by beginner/intermediate students, including myself long ago; while it is part of the self-introduction routine here, it’s more important when asking someone to do something for you. Once you understand its role there, you are more likely to understand what it means during a self-introduction.
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