This question tests your understanding of the subjunctive mood in English, specifically with the phrase "I'd rather" (which is a contraction of "I would rather").

Concept: The phrase "I'd rather" is used to express a preference. When we talk about a preference for a past action that did not happen, we use the past perfect structure.

Rule:

I'd rather + subject + had + past participle

This structure indicates a wish or preference about something that happened (or in this case, did not happen) in the past.

Step-by-Step Analysis:

Step 1: Identify the time context. The sentence mentions "last week" and "this week". The speaker is talking about a past event ("last week") and comparing it to the present ("this week").

Step 2: The structure "I'd rather she..." requires the subjunctive mood to show the speaker's wish about another person's action.

Step 3: Because the wish is about a past action ("last week"), we must use the past perfect form ("had + past participle") to indicate that the action is completed in the past relative to the time of speaking.

Step 4: The past participle of the verb "stay" is "stayed". Therefore, the correct construction is "had stayed".

Final Answer: The most appropriate word to fill in the blank is had stayed.

The complete sentence is: "I'd rather she had stayed away last week rather than this week." This means the speaker wishes her departure had happened last week (a past time) instead of the current week.

Why the other options are wrong:

• stays: This is the simple present tense, used for facts or habits. It does not fit the past context of "last week".
• stayed: This is the simple past tense. While it refers to the past, the structure "I'd rather + subject + past simple" is used to express a preference about a present or future action, not a past one. (e.g., I'd rather she stayed home tomorrow).
• stay: This is the base form of the verb. The structure "I'd rather + base verb" is used when the subject of "rather" is the same as the subject of the following verb (e.g., I'd rather stay home). Here, the subject changes from "I" to "she", so this structure is incorrect.

Related Topic & Theory: The Subjunctive Mood

The subjunctive mood is a verb form used to explore hypothetical or non-real situations, such as wishes, suggestions, or demands. It often appears after certain verbs and expressions like "wish", "suggest", "demand", "it's important that", and "would rather".

Key Structures:

• Present/Future Preference: I'd rather + subject + past simple verb
  Example: I'd rather you called me later.
• Past Preference: I'd rather + subject + had + past participle
  Example: I'd rather you had called me yesterday.

Related Formulae:

Formula for "would rather" with different subjects:
Subject₁ + would rather + Subject₂ + verb form

• If the verb refers to the present or future, use the Simple Past tense.
• If the verb refers to the past, use the Past Perfect (had + past participle).