Both can be used in "quoted speech", but "-다고" is used for adjectives or past markers, 었/았/였". e.g. 예쁘다(adjective, to be pretty) >> 예쁘+다고 / NOT 예쁘라고 먹다(verb, to eat) >> 먹 + 었 + 다고 / NOT 먹었라고 On the other hand, "-라고" is used 1) before "-이다/아니다" when.... 2) verb stem without a last consonant(=받침) 3) verbs ends with the last consonant ' ㄹ' e.g. 꽃+이다(it's a flower) >> 꽃이 + 라고 했다. 꽃이 + 아니다(it's not a flower) >> 꽃이 아니+라고 했다. 가다(no last consonant in the verb stem, '가' ) >> 가 (Go away!) >> 가+라고 했다. 놀다(the verb stem has the last consonant 'ㄹ') >> 놀아 >> 놀 + 라고 했다. \^o^/