Why do you think Claudius go to great lengths to flatter Laertes?

Because Laertes would probably defeat Hamlet in a duel.

Explanation:

In Shakespeare's "Hamlet," King Claudius praises and compliments Laertes to follow his own interest, which is to get rid of Hamlet. Since he cannot kill Hamlet himself because of his love for Gertrude, he adulates Laertes by saying that he has heard that he is extremely good at fighting with a sword. Thus, he encourages Laertes to avenge his father's death so that he can finally execute Hamlet, who suspects Claudius has murdered his own brother to take the throne.