Sorry to disillusion you but they cannot be kept with Goldfish. Goldies are cool water fish and Bettas are Tropicals through and through. They need a temp of 78 to 80 Fahrenheit. Goldies are a dirty fish and bettas are not able to tolerate the amount of waste that they produce without getting finrot or other disease. Goldies require a lot bigger area of tank space....30 gallons for the first one and 10 gallons for each one after that and bettas need the same space as other tropicals. But where a goldfish could be kept in a tank with no heater and be fine with room temperatures, bettas cannot unless you live in the tropics or where the temperature does not fluctuate at night much. They both need filters, there is no such thing as a fish that does not need a filter as they all produce ammonia which converts to nitrites and nitrates. This kills fish and the easiest and most reliable way to deal with it is a filter that has been through the nitrogen cycle. The next best way is to have a heavily planted tank. If you have enough live plants it will take care of this for you but this is for advanced aquarists who really know what they are doing.
Goldfish are cold water fish, but Bettas are tropical fish so I'd agree they wouldn't go well together. Either the goldie would be too hot or the Betta would be too cold and they're not really compatible as Bettas would likely nip at the goldie fins.
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