This question tests your understanding of modal verbs and their use in expressing past possibilities or opportunities that were not taken.

The sentence describes a past situation: "We didn't go out last night." It then presents a hypothetical alternative that was available: "...but we decided to stay at home."

The correct grammatical structure to express a past possibility that was not acted upon is modal verb + have + past participle.

Let's analyze the options:

Step 1: Analyze the context. The sentence is about a past opportunity (last night) that was available but not taken.

Step 2: Evaluate each option.

• could have gone: This means "it was possible for us to go, but we did not." This perfectly fits the context of a missed past opportunity.
• are to go: This is used for formal instructions or arrangements about the future (e.g., "You are to report to the principal."). It does not fit a past context.
• needn't go: This means "it is not necessary to go" and refers to present or future time. It does not fit the past context of the sentence.
• should go: This gives advice or states an obligation for the present or future. It does not refer to a past possibility.

Final Answer: The most appropriate option is could have gone.

The completed sentence is: "We didn't go out last night. We could have gone to the cinema but we decided to stay at home."

Related Topic: Modal Verbs for Past Possibility

Modal verbs like could, might, and would can be combined with "have" and a past participle to talk about the past.

• Could have + past participle: Used to express an ability or opportunity to do something in the past that was not done.
• Might have + past participle: Used to express a possibility in the past that we know didn't happen.
• Would have + past participle: Most commonly used in the main clause of a hypothetical (third conditional) sentence about the past (e.g., "I would have called you if I had known.").

Key Formula

The structure for expressing a past possibility is:

$\mathrm{Modal\; Verb}+\text{ have }+\text{ Past Participle}$