Human-elephant conflict (HEC) is rising across India as habitat loss and expanding human settlements force elephants into closer contact with people. While the ecological drivers of conflict are well-studied, how the news media portrays them remains largely unexplored. This work presents the first large-scale computational analysis of media framing of HEC in India, examining 1,968 full-length news articles consisting of 28,986 sentences, from a major English-language outlet published between January 2022 and September 2025. Using a multi-model sentiment framework that combines long-context transformers, large language models, and a domain-specific Negative Elephant Portrayal Lexicon, we quantify sentiment, extract rationale sentences, and identify linguistic patterns that contribute to negative portrayals of elephants. Our findings reveal a dominance of fear-inducing and aggression-related language. Since the media framing can shape public attitudes toward wildlife and conservation policy, such narratives risk reinforcing public hostility and undermining coexistence efforts. By providing a transparent, scalable methodology and releasing all resources through an anonymized repository, this study highlights how Web-scale text analysis can support responsible wildlife reporting and promote socially beneficial media practices.