He doesn't, really. He wants to see if other people will agree if Reynaldo starts slandering him, and doesn't realize that this plan will hurt Laertes's reputation regardless. He's acting out of curiosity. There's also a wild theory out there that he's specifically trying to get Laertes to kill Reynaldo.
Laertes.
Laertes is the son of Polonius and the brother of Ophelia.
to get a life
In Act 2 of "Hamlet," Reynaldo and Polonius discuss Laertes's behavior while he is studying in France. Polonius instructs Reynaldo to spy on Laertes and gather information about his conduct, emphasizing the importance of finding out whether Laertes is living honorably or engaging in questionable activities. Polonius believes that by observing Laertes's acquaintances and their actions, he can gain insight into his son's reputation and character. This conversation highlights Polonius's controlling nature and his desire to maintain a close watch over his children.
Reynaldo is being sent to spy on Laertes. Polonius authorizes him to accuse Laertes of all kinds of immoral behaviour to find out from his associates whether he is in fact involved in it. The fact that this will tarnish Laertes's reputation in the process does not seem to bother him.
• If Reynaldo follows Polonius’ instruction, he will essentially ruin Laertes’ reputation while trying to gather information.
Polonius, Ophelia, Gertrude, Hamlet, Claudius, and Laertes.
Laertes blamed Prince Hamlet for Polonius' death. He sought revenge against Hamlet for killing his father, believing him to be responsible for the tragedy.
Reynaldo is a spy who is supposed to spy on Laertes.
Ophelia is the daughter of Polonius. She is also the sister to Laertes and potential wife of Prince Hamlet.
Laertes accuses Claudius of killing his father Polonius in Act IV, Scene V.
Polonius is the chief counselor to the king, and also the father of Ophelia and Laertes.