1) as for '인사하기'. Putting '기' at the end of verbs is one way to make a verb a noun. For example, '인사하기' is from '인사하다(to greet)' '먹다(to eat)' can be '먹기'(eating) '공부하다(to study)' can be '공부하기'(studying) The verbs have already changed to nouns, so that they can be a subject in a sentence. like '공부하기는 어렵다'.(studying is not easy.) 2) '이 가방을 사기 보다는 저 가방을 사는게 좋겠어요' As I said already, '사기' is a noun putting '기' as in '사다'(to buy). Also your sentence sounds weird, '이 가방을 사는것보다~' is preferred to '이 가방을 사기보다~' I am not familiar with grammar terms, but I don't think it's a topic particle. This pattern, '~ 하기 보다 ~ 하다' or '~하는것 보다 ~ 하다(하는게 낫다)' is kinda preference, and 'poda' should be 'boda(보다)'. 3)'엄마가 미국에 가 있는 동안 빨래하고 요리도 합니다.' This sentence should be ' 엄마가 미국에 가 있는 동안 빨래도 하고 요리도 했다.' It would have been better if you had asked your question in Korean. Jumping to conclusion, the phrase '가 있는' has nothing to do with 'verb+a/eo/yeo+issda/itta'. As in 'verb+a/eo/yeo+issda/itta', '~가' can be a subject marker, but in your sentence, '가' is just 'has been somewhere' or 'to be out'. for example, As a subject marker,'가' '고양이 한마리가 있다' (there is a cat.) '기회가 있다' (there is an opportunity.)