1. He didn't bother to let me know the fact.

2. He didn't bother letting me know the fact.
(Are both grammatical?)

3. Why didn't you bother to care for the boy?
4. Why didn't you bother caring for the boy?
(Are both grammatical? Can we use 'to do' or 'ing' after the verb 'bother'?)

though sometimes awkward, gerunds and infinitives are usually interchangeable. They both function as nouns, after all.

1. Yes, "He didn't bother to let me know the fact" is grammatical. In this sentence, the verb "bother" is followed by the infinitive form "to let."

2. Yes, "He didn't bother letting me know the fact" is also grammatical. In this sentence, the verb "bother" is followed by the gerund form "letting."

3. Yes, "Why didn't you bother to care for the boy?" is grammatical. In this sentence, the verb "bother" is followed by the infinitive form "to care."

4. Yes, "Why didn't you bother caring for the boy?" is also grammatical. In this sentence, the verb "bother" is followed by the gerund form "caring."

Both the infinitive form and the gerund form can be used after the verb "bother," depending on the desired meaning and context.

1. "He didn't bother to let me know the fact" is grammatically correct. Here, 'to let' is followed by the base form of the verb 'know.'

2. "He didn't bother letting me know the fact" is also grammatically correct. In this case, 'letting' is used as a gerund (verb acting as a noun) after the verb 'bother.'

3. "Why didn't you bother to care for the boy?" is grammatically correct. 'To care' is followed by the base form of the verb 'care.'

4. "Why didn't you bother caring for the boy?" is also grammatically correct. Here, 'caring' is used as a gerund after the verb 'bother.'

In general, both the infinitive form (to + base verb) and the gerund form (verb + ing) can be used after the verb 'bother,' depending on the intended meaning. However, the infinitive form tends to imply an intentional or voluntary action, while the gerund form implies a general or ongoing action.