The sentence "Elle se brosse les dents d'aller se coucher" is meant to convey the sequence of actions or the reason why the action is performed. In this context, it implies that she is brushing her teeth in order to go to bed. The options provided suggest different prepositions that could be used to complete the sentence correctly. Let's briefly examine each:
1. "depuis": This means "since" or "from" in English. It indicates time or origin and would not fit logically into the sentence because it doesn't connect the two actions appropriately.
2. "avait": This is a form of the verb "avoir," meaning "had." It is not a preposition and does not serve to logically connect brushing teeth with going to bed in this sentence.
3. "avec": Meaning "with," this preposition implies accompaniment or association. It suggests doing two actions together but not sequentially or in purpose-based association, which doesn’t align with the intended meaning of the sentence.
4. "derrière": This means "behind" or "afterward." While it indicates sequence physically or in time, this preposition is not typically used to denote the purpose of an action.
The appropriate word to complete the sentence logically would be a preposition like **"avant"** meaning **"before"**, so the sentence would become: "Elle se brosse les dents avant d'aller se coucher." This clearly explains that she brushes her teeth **before** going to bed, showing the correct sequence of the actions.