。
I have to finish this work by next Tuesday.
Literal
Next-week [possessive-の] Tuesday [by-までに] this work [object-を] finish-not-must.
終わらせる is the causative of 終わる ('to end / finish,' intransitive) — 'to cause to end / bring to an end,' functionally equivalent to 終える or 終了させる. Using the causative of an intransitive verb is a regular way to form transitive meaning 'to cause X to happen' where a native transitive partner exists.