When you entrust someone with something, you put an important job — or an important person — in their care, because you believe they will look after it well. You are handing the responsibility over, not just asking a small favour. The usual patterns are entrust someone with a task, or entrust a task to someone. People use it for serious things: a child, a secret, money, the keys, a shop. You would not normally entrust someone with holding your coffee for ten seconds; that is just “can you hold this?” The word feels a little careful and formal. Trust is the feeling that someone is reliable; entrust is the act of actually putting the thing in their hands.