What is the king's main concern when he finds out that Polonius has been killed and why does he decide to turn his "wisest friends"? This is in act 4, scene 1.
Claudius's main concern is that Hamlet has discovered that he, Claudius, killed Hamlet's father. Claudius shows no grief that Polonius has been killed: he apparently doesn't care. Claudius's acute concern is to save his own skin. At this point, Hamlet has shown himself willing and able to kill, so Claudius is concerned to make a move to protect himself. He says, when Gertrude tells him about Polonius, that it would have been him, Claudius, "had he been there" and calls Hamlet a threat to all of them--"to you yourself, to us, to everyone"--and says he must be restrained. He states he will send Hamlet away.
Claudius decides to turn to his wisest friends, tell them what Hamlet has done in killing Polonius, and what he, Claudius, plans to do, because he realizes the situation is dire. More specifically, he wants to get out ahead and do damage control, spinning the story to make Hamlet look bad. He realizes it will not be good for him if the rumor gets out that Hamlet, the late king's son, tried to kill him and only got Polonius by error. As he will say in a later scene, he'd like to throw Hamlet in jail, but doesn't dare do so, because Hamlet is popular with the people. Instead, Claudius will offer an appearance of calm level-headedness in sending Hamlet to England.