Structure
お + Verb[stem]+ する
ご(1) + する[Verb] + する
(1) お, limited to [する]Verbs like: 電話する、勉強する、散歩する
Details
Polite
About お〜する
お〜する comes from a form of polite speech in Japanese called humble speech. This type of speech is used to refer to the speaker and their actions. This is opposed to honorific language, which refers exclusively to the listener and their actions. お〜する is almost exactly the same as お~になる, as in it simply means する.
お〜する is regularly used with words of Japanese origin, while ご~する will be used with words of Chinese origin. The する in this compound will attach directly to the ます stem of almost any verb.
Caution
While the する in this structure will attach directly to the ます stem of verbs, this does not mean that verbs that are already する verbs will become しする. In the case of する verbs, simply change the prefix to ご.
In rare cases, お may be used instead of ご, when attached to words of Chinese origin. There is no specific rule for this, and they must be memorized on a case by case basis.
Thankfully, the words of Chinese origin that use お instead of ご are extremely common verbs. This means that you will be exposed to almost all of them very quickly. All others may be assumed to use ご.
Fun Fact
In the same way that お~になる is a more common variation of なさる (both honorific language), お〜する is a more common variation of いたす (both humble speech).
Related
Examples
--:--
ドアをお閉めします。
I (will) humbly close the door.
お任せします。
I will humbly leave it up to you.
先生の分もお取りします。
I will humbly take the teacher's share.
あなたの分もお作りします。
I will also humbly make your share.
これを彼にお渡ししてください。
Please humbly hand over this to him.
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お〜する – Grammar Discussion
Most Recent Replies (20 in total)

TheA3ther
As others in the thread have stated, I think the two example sentences in this grammar point are incorrect, but for a different reason. The grammar itself is fine, but they don’t use the grammar point that they’re examples of. At least, that’s what I assume is happening.
In the sentence:
ご注文する時はこちらのベルを押してください。
What I believe is happening is that the speaker is referring to the customer’s order, which is why the prefix ご is used. One main use of ご and お is to respectfully describe the subject’s actions. This is why the phrase in the sentence isn’t translated as “humbly order” but rather “order” with the ご prefix for respect.
Same thing happens to the other sentence:
ご登録する方はこちらをクリックしてください。
Here, ご登録 is referring to the actions of those who will register, hence the prefix. Again, this is just what I assume is happening here, would love to be corrected on this if wrong.

rtg142857
I also have Brian’s question. I answered the これを彼にお渡ししてください prompt as これを彼にお渡しになってください because I figured that the 渡す was the action of the listener, and was under the impression that お~になる is used for the listener’s actions; it was marked as wrong, and I’m not sure what makes that wrong/お〜する correct in this case.

casual
I think Brian’s reading is correct.
これを彼にお渡ししてください
in this situation お~する indicates that 彼 is honored, and the listener is comparatively humbled. ください indicates speaker’s politeness to listener. For example, the speaker and the listener are colleagues in the same company, and 彼 is their customer.
彼 > listener >= speaker
With お~になる, in principle it would be the other way around, the listener is honored, and both speaker and 彼 are comparatively humbled.
listener > 彼 ≈ speaker