Japanese to English Art/Literary Poetry & Literature Shakespeare quote from Henry V

Hello, translators! I am currently working on a romance manga from Japanese into English. In one section, a character is quoting a line from Shakespeare's Henry V: 「手を打って取引き成立」. For context, the male character has made a proposition to the female character, wanting her to write down his life's experiences in exchange for letter her use his pool. She seems a bit apprehensive about the deal, because she's afraid of falling for him. Then, the male characters asks, "How about it?", and then says that quote. Following that, the female exclaims that it's from Shakespeare's Henry V.

If I had to roughly translate that sentence, perhaps it would be somewhere along the lines of, "We strike a deal, and the business/deal begins.".

If anyone could help me with a better translation, that would be greatly appreciated. Bonus points if someone knows which quote this might be referring to in Henry V!