Learn Hardcore Korean: i bogoseo jemogeul jedaero sseuja. - Let’s write the title of this report properly.

-자 is the informal propositive ending meaning “let’s (do … together).” It includes both speaker and listener(s). It’s not a command to “you,” but a suggestion/decision for “us.”

Is -자 polite? When should I avoid it?

-자 is casual. Use it with friends/peers or as self-talk. Avoid it with seniors, strangers, or in formal settings. Polite alternatives:

씁시다 (formal “let’s”)
쓰죠 (polite-soft “let’s/shall we”)
쓸까요? (deferential “shall we write…?”)

Why is there no subject like 우리 (“we”)?

Korean often drops obvious subjects. With -자, “we” is built in, so 우리 is optional. You can add it for emphasis: 우리 이 보고서 제목을 제대로 쓰자.

Do I need 의 between 보고서 and 제목? Is 이 보고서 제목 okay?

Both are fine:

이 보고서 제목: common in speech; noun–noun compound (“report title”).
이 보고서의 제목: more explicitly possessive/formal (“the title of this report”). Meaning is the same here.

Why is it 제목을, not 제목를?

The object particle is 을 after a consonant and 를 after a vowel. 제목 ends with a consonant (ㄱ), so 제목을. In pronunciation, the ㄱ links to 을, so it sounds like “제모글.”

What nuance does 제대로 add? How is it different from 잘, 정확히, or 올바르게?

제대로: “properly/as it should be,” often implying previous attempts were sloppy or off-standard.
잘: “well/skillfully.”
정확히: “accurately/precisely.”
올바르게: “correctly/rightly” (according to rules or ethics). So 제목을 제대로 쓰자 = “Let’s do the title properly (format, placement, spelling).”

Does 제대로 usually go with negatives? Is positive use natural here?

It often appears with negatives (e.g., 제대로 못 했어), but positive imperatives/suggestions like 제대로 쓰자 are perfectly natural, especially as a gentle admonition: “Let’s do it right.”

Does 쓰다 here mean “write,” or could it be “use/wear/bitter”? What if I mean “come up with a title”?

Here it means “to write.” Related verbs:

제목을 짓다: to coin/come up with a title.
제목을 정하다: to decide on a title.
제목을 적다: to jot down (copy) a title.
제목을 쓰다: to write/type the title on the document.

How can I make this suggestion more polite or softer?

Formal: 이 보고서의 제목을 제대로 씁시다.
Polite-soft: 이 보고서의 제목을 제대로 쓰죠.
Deferential/asking: 이 보고서의 제목을 제대로 쓸까요?
Very soft/indirect: 이 보고서의 제목을 제대로 쓰는 게 어떨까요?

Can I add 요 and say 쓰자요 to be polite?

Not in standard Korean. Use 씁시다, 쓰죠, or 쓸까요? instead.

Can I move 제대로 elsewhere?

Default placement is before the verb phrase: 이 보고서 제목을 제대로 쓰자. Fronting for emphasis is possible: 제대로 이 보고서 제목을 쓰자 (contrastive “properly, this time…”). Putting 제대로 at the very end is unnatural.

Can I drop the object particle 을?

Yes, in casual speech: 이 보고서 제목 제대로 쓰자. Particle dropping is common when the meaning stays clear. Keeping 을 is a bit clearer and more neutral.

Any quick pronunciation tips?

제목을: links to sound like “제모글.”
쓰자: tense ㅆ; say a strong “s” + ㅡ (eu): “ssŭ-ja.”
보고서: “bo-go-seo,” with 서 as an “uh” sound.

Is 보고서 the only word for “report”? What about 리포트?

Both exist:

보고서: standard/formal term (work, official documents).
리포트: loanword, common in school/university and casual contexts. In formal writing, prefer 보고서.