Learn Hardcore Korean: achim tellebijeon nyuseureul bomyeonseo babeul meogeosseo. - I ate breakfast while watching the morning news on TV.

-면서 attaches to a verb stem to show that two actions happen at the same time. Here, 보면서 means “while watching.” It tells us the speaker was eating breakfast at the same time as they watched the morning TV news.

Why is the verb 먹었어 in an informal style, and how would I make it polite?

먹었어 is the banmal (informal) past tense of 먹다. To make it polite (jondaetmal), change it to 먹었어요. The first verb’s connective form stays the same, so you’d say:
• Informal: 아침 텔레비전 뉴스를 보면서 밥을 먹었어.
• Polite: 아침 텔레비전 뉴스를 보면서 밥을 먹었어요.

Why isn’t there a particle after 아침? Shouldn’t it be 아침에?

Here, 아침 is part of the compound noun 아침 텔레비전 뉴스 (“morning TV news”), so no particle is needed. If you want to say “in the morning” as an adverbial time phrase, you use 아침에 and treat 텔레비전 뉴스를 separately:
아침에 텔레비전 뉴스를 보면서 밥을 먹었어.

Where is the subject of this sentence? Why is it omitted?

In Korean, subjects (and objects) are often dropped when they’re clear from context. Here the implied subject is “I” (나/저). If you need to emphasize it, you can say:
나는 아침 텔레비전 뉴스를 보면서 밥을 먹었어.

Could I swap the order of the actions, like 밥을 먹으면서 아침 텔레비전 뉴스를 봤어? Would that change the nuance?

Yes. Both sentences describe two simultaneous actions, but the main clause (the part not in -면서) feels more in focus.
• 보면서 밥을 먹었어 → Focus on eating while watching.
• 밥을 먹으면서 뉴스를 봤어 → Focus on watching while eating.

What’s the difference between connecting verbs with -면서 vs. -고?

• -면서 specifically stresses that the actions overlap exactly (“while doing X, Y”).
• -고 simply lists or sequences actions (“X and then Y” or sometimes simultaneously, but less emphatic).
Example:
– 뉴스를 보면서 밥을 먹었어. (I was eating breakfast while watching the news.)
– 뉴스를 보고 밥을 먹었어. (I watched the news and ate breakfast—less clear if at the same exact moment.)

Why is it 보면서 and not 봐면서 or 보하면서?

For most Korean verbs, you drop -다 and add -으면서 or -면서 (if the stem ends in a vowel). 보다 → stem 보 → 보면서.
You use -하면서 with 하다 verbs or Sino-Korean nouns:
• 공부하다 → 공부하면서
• 청소하다 → 청소하면서

Why are the particles -를 on 뉴스 and -을 on 밥 necessary? I sometimes see them dropped.

When a verb takes an object, you normally mark it with -을/를:
• 뉴스를 보다
• 밥을 먹다
In very casual spoken Korean, particles can be dropped if the meaning is obvious:
• 뉴스 보면서 밥 먹었어.
But in writing or more careful speech, it’s best to include -을/를 for clarity.