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
• If Reynaldo follows Polonius’ instruction, he will essentially ruin Laertes’ reputation while trying to gather information.
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.
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.
In Hamlet, Reynaldo is obligated to follow the instructions given to him by Polonius because Polonius is his superior Polonius sent Reynaldo to Paris so he could spy on Laertes for him..