Which word means to think carefully and deeply about something?

The word that means to think carefully and deeply about something is "ruminate." This term derives from the process of cows chewing their cud, which is a metaphor for pondering or contemplating an idea thoroughly. It captures the essence of thoughtful consideration, often implying a thorough and reflective approach to thinking.

In contrast, the other options do not convey this deep level of contemplation. "Inveterate" refers to a long-established and unlikely to change habit or condition. "Quell" means to suppress or put an end to something, often in the context of emotions or unrest. "Trenchant" describes something that is incisive, sharp, or effective, especially in expression, but it does not specifically relate to the act of deep thinking. Hence, "ruminate" is the most fitting choice for this definition.