"The mother seems busy."
Translation:어머니는 바쁘게 보인다.
바빠 보이다 sounds more natural. When you want to say something looks (in the sense of "seems") you conjugate the verb and add 보이다. 학생은 피곤해 보여요 - the student looks tired.
When you want to say seems (in more figurative ways) you have a different structure. You add ~은/는/을 것 to the verb (for past, present, or future tense) and 같다. 이 보드게임은 어려운 것 같아요. - This board game seems difficult.
엄마가 밖에 간 것 같아요 - it seems like mom went out. / 엄마가 밖에 가는 것 같아요 - it seems like mom is going out. / 엄마가 밖에 갈 것 같아요 - it seems like mom will go out.