sukjereul da kkeutnaego naseo jibeseo swieoyo.

숙제 means homework. The particle -를 marks it as the direct object of the verb 끝내다 (to finish).
So 숙제를 끝내다 = to finish homework.

다 is an adverb meaning all / completely. It emphasizes that the homework is finished entirely.

• 숙제를 끝내고 = after finishing the homework
• 숙제를 다 끝내고 = after finishing all the homework

It’s a common sequence pattern:

• 끝내- (verb stem of 끝내다, to finish)
• -고 = and/then, often used to link actions
• 나서 (from 나서다, “to come out/after doing”) forms the fixed pattern -고 나서 meaning after doing (something), then...

So 숙제를 다 끝내고 나서 = after finishing all the homework (and then).

They’re very similar: both mean after doing.

• -고 나서: very common in spoken Korean; often feels like do A, and then (immediately) do B
• -(으)ㄴ 후에: a bit more neutral/formal; more like after A (at some point), B

In this sentence, both work:

• 숙제를 다 끝내고 나서 집에서 쉬어요.
• 숙제를 다 끝낸 후에 집에서 쉬어요.

Both can translate to at home, but they have different typical uses:

• 집에 focuses on destination/existence (go to / be at)
• 집에서 focuses on location where an action happens

Because 쉬어요 (I rest) is an action, Korean commonly uses -에서:

• 집에서 쉬어요 = I rest at home

쉬어요 is the polite present form of 쉬다 (to rest). Conjugation:

• 쉬다 → stem 쉬-
• add -어요 → 쉬어요

It’s the common polite style used in everyday conversation.

Formally, 쉬어요 is present tense, but in Korean the present tense often covers:

• habit/routine: I (usually) rest at home after finishing homework
• near-future plan (context-dependent): I’ll rest at home after I finish homework

If you want to clearly mark future intention, you might hear:

• 쉴 거예요 = I’m going to rest

Korean often omits the subject when it’s obvious from context. The sentence is still complete without it.
If you want to include it:

• 저는 숙제를 다 끝내고 나서 집에서 쉬어요. (polite)
• 나는 숙제를 다 끝내고 나서 집에서 쉬어. (casual)

They’re different verb types:

• 끝내다 = to finish (something) (transitive; takes an object)
  → 숙제를 끝내다 (finish homework)
• 끝나다 = to be finished / to end (intransitive; no object)

So you generally wouldn’t say 숙제를 끝나고. But you could rephrase:

• 숙제가 다 끝나고 나서 집에서 쉬어요. = After the homework is all done, I rest at home.

Yes, but the meaning shifts slightly:

• -고 나서 focuses on sequence: after A, then B
• -아서/어서 often implies reason or natural result: because A / so A, (therefore) B

So:

• 숙제를 다 끝내고 나서 집에서 쉬어요. = sequence (A then B)
• 숙제를 다 끝내서 집에서 쉬어요. = “I finished it, so I rest at home” (result/reason nuance)

Standard spacing is 끝내고 나서 (separated). It’s treated as verb + connector (-고) + 나서. In casual typing you might see 끝내고나서, but the spaced form is the standard.

Same content, different politeness:

• More formal polite: 숙제를 다 끝내고 나서 집에서 쉽니다.
• Casual: 숙제 다 끝내고 나서 집에서 쉬어.
• Casual with 난/나는: 난 숙제 다 끝내고 나서 집에서 쉬어.