Can you please explain act 4 scene 5 in detail? What is the significance of the King's conversation with Ophelia?
Ophelia comes to speak with Gertrude, and it is very clear that she has "gone mad," as they say. She enters, singing a song about death: "He is dead and gone, lady, / He is dead and gone" (4.5.34-35). It is likely that this is in response to her father's death, a death which came at the hands of her ex-lover. Then Claudius joins them. She begins to sing another song, this one about young lovers. The young man "Let in the maid, that out a maid/ Never departed more" (4.5.59-60). In other words, his lover entered the room a virgin but she did not leave as one. However, she said, "'Before you tumbled me,/ You promised me to wed.'" (4.5.67-68). So, he had promised to marry her before they slept together, but now that they have had sex, he no longer wishes to marry her. It seems most likely that she is thinking about Hamlet now. She returns to the subject of her father, speaking to people who aren't really there, and she leaves.
The king communicates his concern that the Danish people are suspicious of him because Polonius was buried so quickly and without ceremony; people suspect him of wrongdoing because the burial was not public. He also says that he's heard Laertes has returned from France in secret because he, too, is suspicious of Claudius. Suddenly, they hear a terrible noise outside and a messenger arrives to tell them that Laertes is at the gates with a huge crowd of supporters who want him to be king. When Laertes enters, he confronts Claudius and tells him that he must avenge his father's death or else he cannot truly count himself Polonius's son.
As he and Claudius talk, Ophelia wanders back in, and Laertes can tell instantly that she has lost her wits. She sings more and offers each of them flowers (although it is likely that she only mimes handing out the flowers and doesn't actually hold any). Each flower symbolizes something different. To Gertrude, she gives fennel (flattery, deceit -- she has been susceptible to both from Claudius), columbines (ingratitude, adultery -- she was not loyal to her dead husband when she married so quickly after he passed away), and rue (regret, repentance -- she now regrets her behavior). To Claudius, Ophelia gives daisies (unrequited love) and rue that she says he must wear "with a difference" (4.5.207). She also keeps rue for herself. The flowers that she gives to each indicate that, on some level, even her distracted mind is able to suss out some truth. To Claudius, especially, she seems to indicate that his love for Gertrude is no longer reciprocated and that his regret is of a different kind than either Gertrude's or her own; Ophelia likely regrets ending her relationship with Hamlet, Gertrude likely regrets her behavior following the death of her first husband, but Claudius's regret is of a much darker kind because he has committed murder. The flowers she gives him seem to indicate that she knows his secret. Then she sings another song about death and exits. In the end, Claudius and Laertes agree to speak more about Polonius's death and funeral.