Final answer:
The correct sentence is: They promised to be quiet in the car.
Step-by-step explanation:
The correct sentence is: They promised to be quiet in the car.
In this sentence, 'to be quiet' is an infinitive phrase acting as the direct object of the verb 'promised'. The base form of the verb 'be' is used together with the infinitive marker 'to' to indicate an action or state that is promised or agreed upon.
For example, if someone says 'I promise to help you,' the infinitive phrase 'to help you' is used to show the action being promised. Similarly, in the sentence 'They promised to be quiet in the car,' the infinitive phrase 'to be quiet' shows the action of being quiet that is promised by 'they'.
The correct sentence between "They promised being quiet in the car." and "They promised to be quiet in the car." is the latter. The structure 'promise + to infinitive' is the correct form when indicating that someone has given their word to do something. Therefore, the sentence should be "They promised to be quiet in the car."