This question is about converting direct speech into reported speech. The direct speech is: "Why don't we go out for dinner?" This is a suggestion. In reported speech, we change the pronouns and verb tenses to reflect that the words are being reported, not spoken directly.

Step 1: Identify the Type of Sentence
The sentence "Why don't we...?" is a common way to make a suggestion. The reporting verb for a suggestion is "suggested".

Step 2: Change the Pronoun
The pronoun "we" refers to the speaker (his wife) and the listener (her husband). In reported speech, this becomes "they".

Step 3: Change the Verb
The verb "go" needs to be changed. After the verb "suggest", we often use the structure "that + subject + should + base verb". The word "should" is commonly used to report suggestions, though it can sometimes be omitted.

Step 4: Form the Reported Sentence
Following these rules, the direct suggestion becomes: they should go out for dinner.

Final Answer: The correct option is "They should go out for dinner".

The other options are incorrect:

• "They would go..." implies a future conditional, not a suggestion.
• "We will go..." and "We'd like to go..." do not change the pronoun "we" to "they".

Related Topic: Reported Speech

Reported speech (or indirect speech) is used to tell someone what another person said. The main changes involve:

• Pronouns: They change according to the perspective of the reporter (e.g., I → he/she, we → they).
• Verb Tenses: They often "shift back" in time (e.g., present simple → past simple).
• Time and Place Words: Words like 'now', 'today', and 'here' change to 'then', 'that day', and 'there'.

Key Formulae/Rules

For Suggestions:
Direct: "Why don't we...?" / "Let's..."
Reported: Subject + suggested + (that) + subject + (should) + base verb
Example: "Let's leave." → He suggested that they should leave.