This question tests your understanding of verb patterns after the verb "regret" and the correct tense to express past actions.

Step 1: Understand the meaning of "regret"

The verb "regret" means to feel sorry about something. It can be followed by either a gerund (verb + ing) or an infinitive (to + verb), but with different meanings.

• Regret + gerund ($\mathrm{not}\mathrm{having}\mathrm{bought}$) is used to express regret about a past action.
• Regret + infinitive ($\mathrm{not}\mathrm{to}\mathrm{buy}$) is used to formally announce bad news, often in writing (e.g., "We regret to inform you...").

Step 2: Analyze the context

The sentence is "I greatly regret _____ those boots when they were so cheap." The phrase "when they were so cheap" clearly indicates that the opportunity to buy the boots was in the past. Therefore, the speaker is expressing sorrow about a past action (or inaction) that did not happen.

Step 3: Evaluate the options

• not to buy: This is an infinitive. It is incorrect because the context is about a past regret, not announcing news.
• not to have bought: This is a perfect infinitive. While it refers to a past action, it is not the standard pattern used after "regret" for this meaning. It is grammatically awkward here.
• didn't buy: This is a finite verb in a past tense clause. The verb "regret" cannot be directly followed by a clause starting with a finite verb like this. The correct structure would require a conjunction, e.g., "regret that I didn't buy".
• not having bought: This is a negative gerund (or gerund phrase). It perfectly follows the rule for "regret + gerund" to express a past action that one is sorry about. The perfect aspect "having bought" emphasizes that the action (or inaction) is completed in the past.

Final Answer: The most appropriate word to fill in the blank is not having bought.

The complete correct sentence is: I greatly regret not having bought those boots when they were so cheap.

Related Topic: Verb Patterns

Many verbs in English are followed by specific structures, like gerunds or infinitives. Knowing these patterns is crucial for grammatical accuracy. Other verbs that are typically followed by a gerund include: enjoy, avoid, consider, and suggest.

Related Formulae/Structures

Regret + Gerund: Used for past actions.
Formula: Subject + regret + (not) + verb-ing
Example: I regret not studying harder for the exam.

Regret + Infinitive: Used to formally announce bad news.
Formula: Subject + regret + to + verb
Example: We regret to inform you that your application was unsuccessful.