Last night in class, we learnt about Verb + 자고 하다 and Verb+(으)라고 하다. As usual, I got confused. From my 선새님’s explanation, this is what I understand:

Verb + 라고 하다
– Someone said someone else has to do something.
선생님이 오늘 까지 숙제를 내라고 하셨어요. / The teache said that I have to hand in the homework by today.

Verb + 자고 하다
– Someone suggested to someone else to do something.
시간 되면 커피를 한잔 마시자고 해요. / (He said) Let’s drink a cup of coffee if you have the time.

So I asked my friend if “-라고 하다” is the same as “-라” and if “-자고 하다” is the same as “-자”, except that one is a direct speech (active), and the other is quotation (passive).

To illustrate my concern,
우리 식당에 가자. / Let’s go to the restaurant.
우리 식당에 가자고 했다. / (He/ She/ They said) let’s go to the restaurant.

She (my friend) told me is the same thing, and told me not to think too much about it. It’s all the same (all implying a suggestion to do something): -자고 하다 & -자 & (ㅂ)시다.

After much deliberation we concluded that I’m right.

I am… right?