Formal:
* Examine: To inspect or study something closely.
* Scrutinize: To examine something very carefully and thoroughly.
* Probe: To explore or inquire into something in detail, often with a view to uncovering hidden information.
* Research: To study a subject thoroughly and systematically.
* Explore: To examine or discuss a subject or issue in detail.
Informal:
* Look into: To investigate something.
* Check out: To examine or investigate something.
* Dig into: To investigate something thoroughly.
* Get to the bottom of: To find out the truth about something.
* Uncover: To reveal something hidden or unknown.
The best synonym to use will depend on the specific context and the desired level of formality.