"Jihoon! You forgot the trash!"
Translation:지훈아! 쓰레기를 잊어버렸다!
From the English phrase alone, 잊어버렸어요 should be accepted as well as 잊어버렸다.
Indeed. I also think there are contexts where the -요 would be used with this sentence.
But it is good to know that the -다 form seems to be commonly used to express an exclamation.
Like when your friends are saying
맛있다! (or 맛있겠다)
whenever a new dish arrives in the restaurant.
Implying, that looks delicious!
I'm saying this based on personal experience, couldn't find any outside online source corroborating this. So take it with a grain of salt.
the 아 after 지훈 isn't a conjugation thing. it's more like trying to get their attention. or letting them know that you are talking to them. in english it's kinda like "hey 지훈!"
You can drop object markers like 를 there is no rule that you HAVE to put them in the sentence.
From what I gather, they are not always used in spoken language but are required in proper written language.
All the "official" classes I've taken always emphasize the importance of getting used to using them rather than omitting them.