There is an example sentence in the grammar points that I don't quite understand.
彼はエンジンをかけっぱなしにしておいた。
what does おいた means here?
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Javizy wrote:「おく」by itself means 'to place', but as an auxiliary verb in the form「~ておく」, it indicates that the action is done for some future purpose or convenience. So in「彼はエンジンをかけっぱなしにしておいた」,「おいた」adds the nuance that he left the engine running in preparation for something (probably so they could drive off without delay). With transitive verbs, and in the past tense, it's very similar to but more commonly used than「~てある」, which also indicates that an action was done for some specific purpose.
「~ておく」can also be used when asking for a favour. In one of the last beginner lessons (where this was covered, actually), there was an example like「資料を会議室に置いておいてください」'please put the documents in the meeting room (in advance for the meeting)'. It's one of the many 'nuancey' elements of Japanese that are rarely translated into English, which you really need to familiarise yourself with until they become second nature.