Both sentenses are right. 1.~는거.(~는것) ~는거 usually express like behavior,action. So action,behavior + 는거.(는것) Ex) 나는 노래 듣는거 좋아해. 나는 노래 듣는것을 좋아해. 두 문장 다 가능합니다. 2.~는지 ~는지 has vague doubt. So ~는지 extend the following paragraph's fact or judgement. Ex) 너가 거기 있는지 몰랐어. 내가 무슨 행동을 했는지 모르겠어.

는지 means the fact of something. 숙제있는지 몰랐다---> I didn't know the fact that I had homework to do. 그가 춤췄는지 몰랐다--->I didn't know that he danced at the party.

감사합니다~

춤추는 거---> To dance, 밥먹는 거--->To eat, 걷는 거--->To walk, 숙제하는 거--->To do your homework, so 는 거 means to do a certain activity