I think "requires the comma to go inside" should be "requires the comma go inside", like "I saw him go to school", right?

AmE punctuation requires the comma to go inside the quotation marks, and I think it's the same in BrE.
Hello Jobb

The verb 'require' needs a 'to-infinitive'. The verbs that allows to be in the construct of V+O+a bare infinitive are very limited; see, hear, feel, listen to , look at, notice, observe, watch, make, have, let and help.

"need" in non-assertive contexts can be used as a modal, i.e., followed by a bare infinitive. Paco gave examples of non-modals that can take a bare infinitive; all modals take the bare infinitive (including non-assertive "need", optionally), of course.

Hope that helps.

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I think I need to check out my grammar book again.
I need hardly tell you how serious the matter is!


(So I can express " I think I need to check out my grammar book again" as "I think I need check out my grammar book again"? Wow, a bit weird! )

Well, just behind "need hardly"! Not 'need"! Emotion: big smile
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 CalifJim's reply was promoted to an answer.