Short Answer 
Claudius showcases his manipulative nature by strategically appealing to Laertes’s emotions, using flattery to gain his loyalty and inciting feelings of revenge for Polonius’s death. This calculated approach aims to orchestrate a duel with Hamlet, demonstrating his ability to exploit emotional triggers for his ulterior motives.
Step 1: Recognizing Claudius’s Manipulative Nature
In this excerpt, Claudius demonstrates his manipulative abilities as he seeks to eliminate Hamlet. His approach is calculated and strategic, appealing to Laertes’s emotions and sense of duty. By acknowledging Laertes’s talents, he sets the stage for his ulterior motive against Hamlet.
Step 2: Flattering Laertes to Gain His Loyalty
Claudius employs flattery as a tool to sway Laertes in his favor. He emphasizes Laertes’s exceptional fencing skills, suggesting that even Hamlet admires him. This not only boosts Laertes’s ego but also plants the seed of rivalry, creating a desire within Laertes to prove himself against Hamlet.
Step 3: Inciting Revenge for Polonius’s Death
Claudius further manipulates Laertes by reminding him of his father, Polonius, and the wrongs done by Hamlet. He poses a crucial question about Laertes’s feelings regarding revenge, tapping into his anger and grief. Claudius’s ultimate goal is to orchestrate a duel, showcasing how he expertly uses emotional triggers to ignite conflict.