The Hamlet Project

Why though? Are the people of Denmark going to rise up and protest the murder of a high state official by a member of the royal family? Seems to me that Royals have people killed all the time and no one says a thing about it. It is one of the distinct advantages of being a royal. Claudius is making it sound like it’s going to be quite a difficult PR job to excuse Hamlet from the murder. But would it? I mean, Hamlet is beloved by the people. The laws are pretty loose for royals. He’s already got a great madness excuse in the making.
Who is he trying to impress here? Gertrude?
Why not just turn the authorities loose on him?
(Well, aside from the fact that his way will actually kill Hamlet and trying his nephew/stepson in a court might reveal some facts that the king might not want revealed.)