How do Claudius and Gertrude react to Polonius's theory regarding Hamlet and Ophelia?

Polonius appears convinced that Hamlet is mad, driven insane by his love for Ophelia. His general behavior around the court and his treatment of Ophelia appear to suggest so. But Claudius is far from convinced. He senses that Hamlet is feigning madness for some ulterior motive. But in any case, if Hamlet really is mad, Claudius wants to make sure what the cause of it is. He knows that Hamlet resents him for marrying his mother, but if he's mad because of Claudius, then that makes him a potential danger. For her part, Gertrude believes that Hamlet is mad because of his father's death and her subsequent remarriage. If this is the case, then Claudius is potentially in deep trouble and so needs to make a move.
Unfortunately for Claudius, the contrived scene he arranges with Polonius between Hamlet and Ophelia proves inconclusive, to say the least. Hamlet knows he's being watched, so as well as lashing out at Ophelia ("Get thee to a nunnery!") he also rails against Claudius, but without mentioning him by name:

"I say, we will have no more marriages. Those that are married already, all but one, shall live." (Act III Scene I).

Claudius's plan has backfired spectacularly and he now finds himself in a worse position than before. There's nothing for it. He's going to settle accounts with Hamlet once and for all by sending him off to England, where he intends for him to be assassinated.