Gertrude and Horatio sadly discuss the madness that has taken over Ophelia since Polonius was killed. Ophelia enters, singing mournful songs about her father.
Hamlet's madness is feigned. Ophelia's is real. As a woman, Ophelia can't act, so she goes mad.
Claudius enters. Ophelia's madness upsets and unnerves him. Ophelia's songs change topic, and focus on maids who are seduced. She exits with the comment that her brother shall know of her father's death. Horatio follows her.
Do Ophelia's songs about seduced maids indicate that she had a sexual relationship with Hamlet? This is an unresolved question in the play.
Claudius mentions that the commoners are also angry about Polonius's death, and that Laertes has secretly sailed back to Denmark. A messenger rushes in with news that Laertes is actually marching toward the castle at the head of a mob chanting "Laertes king!"
Contrast with Hamlet: as soon as Laertes hears of his father's murder, he returns to Denmark and nearly starts a revolution!
Gertrude exclaims that the mob and Laertes are blaming the wrong person for the death of Polonius.
Ironic that Gertrude defends the man who killed her husband.
Laertes bursts into the room. Claudius asks for calm. Laertes retorts that to be calm would make him a bastard, that he would dare damnation just to get revenge for the death of his father. Claudius admits that Polonius is dead. Gertrude adds that Claudius did not kill him.
Another point of comparison with Hamlet in terms of willingness to act to get revenge.
Ophelia enters. She is clearly insane, singing songs, speaking in riddles, and handing out flowers (perhaps imaginary): rosemary and pansies to Laertes; fennel and columbines to Gertrude; rue and daisies to Claudius. Laertes demands vengeance for her madness. Ophelia exits, wishing God's blessing on everyone.
The flowers held symbolic meaning in Shakespeare's time. Rosemary for remembrance. Pansies for thoughts. Fennel for flattery. Columbines for infidelity. Daisies for seduction.
Claudius asks Laertes to let him explain what happened to Polonius, and promises to hand over the crown to Laertes if, after the explanation, his actions still strike Laertes as unjust.
Laertes acts without thinking. Claudius can manipulate those who don't think and turn their actions to his own advantage.