How do petta and katniss gain the support of hamitch?

Peeta and Katniss gain the support of Haymitch by showcasing their strong survival instincts and their willingness to work together as a team during The Hunger Games. Their authenticity and resilience resonate with him, prompting him to see their potential as victors. Additionally, their ability to attract public attention and sympathy helps Haymitch recognize the strategic advantage of investing in their success. Ultimately, their determination to survive and protect each other earns Haymitch's respect and commitment to support them.