The phrase "ask intrusively" can have an unpleasant connotation, as it suggests someone asking questions in a nosy or invasive manner. Luckily, there are several synonyms that convey the same meaning without the negative connotations. Words like "probe," "inquire," and "interrogate" all depict a questioning tone, but in a more neutral or even clinical way. Similarly, "question extensively" or "pursue answers" convey an investigative tone that is not necessarily intrusive. Ultimately, the choice of synonym will depend on the intended tone and purpose of the questioning, but there are plenty of options to avoid unintentionally offending or alienating others.