ずにはおかない is formed by combining the classical negative auxiliary ず with the particles に and は, followed by おかない, the negative form of the verb 措(お)く ('to leave something as it is'). This expression conveys that an action will definitely be carried out, or that a certain outcome is inevitable.

This is a rather formal structure. Its literal sense is 'to not leave a situation alone without doing (A)', which implies a strong determination to see the action through to completion, as suggested by its common English translations.

Since this grammar pattern uses the negative ず ending, it always attaches to the stem of a verb.

• あの映画(えいが)は大人気(だいにんき)だから、何(なに)かの賞(しょう)を取(と)らずにはおかないだろう。
  That movie is so popular, I'm sure it's bound to win some kind of award.
• 人気(にんき)アイドルのスキャンダルだから、世間(せけん)は騒(さわ)がずにはおかない。
  Since it's a scandal involving a popular idol, the public will undoubtedly make a fuss about it.

It is also possible to replace ず with ないでは, forming ないではおかない. This alternative has exactly the same meaning and usage.

• 何度(なんど)も同(おな)じミスをしたからには、部長(ぶちょう)も彼(かれ)を叱(しか)らないではおかないだろう。
  Seeing as he's made the same mistake so many times, the department manager is sure to give him a talking-to.
• 他人(たにん)に迷惑(めいわく)をかけたのだから、子供(こども)には謝(あやま)らせないではおかない。
  Since my child caused trouble for someone else, I will make sure they apologize.