The analysis of students' emotions and behaviors is crucial for enhancing learning outcomes and personalizing educational experiences. Traditional methods often rely on intrusive visual and physiological data collection, posing privacy concerns and scalability issues. This paper proposes a novel method leveraging large language models (LLMs) and prompt engineering to analyze textual data from students. Our approach utilizes tailored prompts to guide LLMs in detecting emotional and engagement states, providing a non-intrusive and scalable solution. We conducted experiments using Qwen, ChatGPT, Claude2, and GPT-4, comparing our method against baseline models and chain-of-thought (CoT) prompting. Results demonstrate that our method significantly outperforms the baselines in both accuracy and contextual understanding. This study highlights the potential of LLMs combined with prompt engineering to offer practical and effective tools for educational emotion and behavior analysis.