1

I'm having some trouble finding out what おけない means in this sentence, この程度の条件を達成できなければ安心して娘たちを任せてはおけないよ.

In the second clause of this ば sentence, the speaker is saying something about his peace of mind and entrusting his daughters and their studies to their tutor (the person with whom he is speaking to). I'm a bit confused what おけない means here though. Entrusting his daughters is the topic of this clause but I don't know what this verb is saying about the topic. My jisho and google searches have come up empty. If anyone could help me out here, I would really appreciate it.

3

任せてはおけない is an emphatic form of 任せておけない, where おけない is a negative of おける (potential of おく).

There are other questions on the site dealing with ~ておく so I'll skip the explanation but the general meaning of the sentence seems to be "I can't very well leave [my] daughters in [your] care with such conditions".

Step by step breakdown:

任せておく - leave in care/trust

任せておける - able to leave in care/trust

任せておけない - unable/cannot leave in care/trust

任せてはおけない - definitely cannot leave in care/trust

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