Final answer:
Methane (CH4) is the greenhouse gas produced by livestock farming and landfill decomposition, with a major influence on global warming.
Step-by-step explanation:
The greenhouse gas generated by livestock farming and the decomposition of solid waste in landfills is methane (CH₄). Livestock such as cows produce methane through their digestive processes in a phenomenon known as enteric fermentation.
Additionally, when organic waste decomposes anaerobically (without oxygen) in landfills, it also produces methane. This potent greenhouse gas has a significant impact on global warming, having approximately 25 times the warming potential of carbon dioxide over a 100-year period.