Enter King, then Rosencrantz, Hamlet, and Guildenstern:
We see the King and two or three attendants. The attendants say nothing, but they're important. They shouldn't be servants or soldiers, but gentlemen and courtiers, because the King is telling them the official position. The official position is that the King is taking firm and reasonable action. He has sent people to find both Hamlet and the body of Polonius. He cannot prosecute Hamlet under the law because "He's loved of the distracted multitude" (4.3.4). Furthermore, it must appear that Hamlet's trip to England is the result of "deliberate pause," that is, careful consideration. The King's attendants are supposed to understand all of this, and they are also supposed to understand that if it appears that the King is acting in desperation, it's because he has no choice: "diseases desperate grown / By desperate appliance are relieved, / Or not at all" (4.3.9-11).
Now Rosencrantz shows up with the news that although they have Hamlet, he won't tell them where Polonius' body is. The King orders Hamlet brought in, and Rosencrantz calls to Guildenstern to "bring in the lord." Rosencrantz has said that Hamlet is "guarded," so in performance, Hamlet often comes in with soldiers who have their swords drawn on him, so he can't get away.
Hamlet immediately begins mocking the King every chance he gets. The first time the King asks where Polonius is, Hamlet tells him that Polonius is "at supper," "Not where he eats, but where he is eaten" (4.3.19). The second time the King asks, Hamlet answers that Polonius is in heaven and that the King can "send hither to see: if your messenger find him not there, seek him i' the other place yourself" (4.3.33-35). Hamlet's joke is that he's just told the King to go to hell. Finally, Hamlet relents a little and says that if Polonius isn't found within a month, he'll begin to stink, and "you shall nose him as you go up the stairs into the lobby." Immediately the King sends some attendants to check out the stairs. Apparently the attendants run or jog, because Hamlet shouts after them "He will stay till you come." In other words, "There's no need to hurry, because he's not going anywhere, being dead and all."
Now the King moves on to the second order of business--getting Hamlet off to England. He tells Hamlet that he's sending him to England for his own safety and that he must go with "fiery quickness." Hamlet answers "good," and the King responds, "So is it, if thou knew'st our purposes," meaning that he's doing it all for Hamlet's own good. Hamlet answers "I see a cherub that sees them" (4.3.48), meaning that he knows that heaven knows that the King's purposes are not good at all. The modern equivalent of this is "Yeah, right."
Just before he goes, Hamlet takes one last poke at the King, by calling him "dear mother." The King responds, "Thy loving father, Hamlet," and Hamlet answers with this: "My mother: father and mother is man and wife; man and wife is one flesh; and so, my mother. This conundrum sounds mad, in a witty way, but it also states what is driving Hamlet "mad": he can't think of the mother he loves without thinking of her as a part of the "father" he hates.
Exeunt all but the King: