Methane and nitrous oxide are both very "strong" greenhouse gasses. And they both break down over a few years - but part of the resulting gasses after that include carbon dioxide.
Nitrous oxide comes from water soluble nitrogen fertilizers (that we make from natural gas i.e. methane) that washes out of the soil with the first water/rain.
So, if methane is released directly into the air, or if it burned (combined with oxygen) - either way, we get more greenhouse gasses. If we get the methane from landfills or from sewage or farm waste, then that carbon came very recently from the air, and so it does not *add* to the carbon dioxide level in the air, over time.
It's only when the carbon comes from underground, from where it had been sequestered for millions and millions of years that it adds to the level of carbon dioxide in the air, that it becomes a problem.
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