One of the more polite ways to express disinterest or unconcern is to use the phrase "not to care a hang." However, there are many other synonyms for this phrase that one can use to convey the same sentiment in different ways. These synonyms include not giving a fig, not giving a damn, not giving a toss, not giving a hoot, and not giving a rap. Each of these phrases suggests varying degrees of indifference, from mild apathy to complete disinterest. Whichever phrase you choose, be sure to use it according to the intended context and tone of your conversation, to ensure effective communication.