There are many synonyms for the phrase "care to," which is commonly used to politely invite someone to do something. Some options include "would you like to," "do you want to," "are you interested in," and "shall we." Each of these phrases conveys a similar meaning to "care to," but may be more appropriate in certain contexts or with different levels of formality. Other alternatives might include "feel free to," "be my guest," or "let's." Choosing the right synonym for "care to" can help you to communicate more clearly and effectively, and could even impact the response you receive from the person you are addressing.