I'm learning English, but my English is very bad. I have a question about this sentence: The principal approached me and asked me that why had I not brought the book? When I translate the sentence I don't understand that. Because I searched on the internet about "had I not + V_pII/V_ed" structure, I see that it likes "If I had not + V_pII/V_ed" structure. It is a Condition Clause, and I can't translate it. I'm so confused. Can anyone help me? Thanks!
As it stands, the sentence is wrong. The inclusion of "that" is the most glaring error, but the reporting of a question is not a question and should be ordered differently. As @deadrat says in comments, English interrogative clauses have a different word order from English declarative clauses, and indirect speech counts as the latter.
The principal approached me and asked me why I had not brought the book. - no question mark - would be a more natural phrasing.
I've heard people using a semi-quoting style, where the question is not quoted as speech but rephrased to match the persons in the sentence. I'm not quite sure how to punctuate it, but it sounds like: The principal approached me and asked, why had I not brought the book.
Then of course there is full-on reported speech: The principal approached me and asked, "Why have you not brought the book?" - and finally in this version we get the question mark.
As you can see in this final version, the word order that you were concerned about is legitimate, and typical of question phrasing with the verb following the question word, This also resembles the simple confirmation question set-up: Have you brought the book?
The position of "not" adjacent to the verb that it is negating is also valid, but it wouldn't be unusual to see (or hear) "Why haven't you brought the book?", the standard contraction "haven't" implicitly drawing the "not" out of position (for contemporary usage). "Why have not you brought the book?" would be wrong, although it may seen in some historical usage, as @1006a points out; for example in Jane Austen's writings:
"Why had not Miss Woodhouse revived the former good old days of the room?"(Emma, Ch. 6)