Converting a request in direct speech to indirect speech generally follows the pattern "reporting verb (asked/begged/requested) + object + to/not to + base verb", so let's test each option against that structure and against what the original sentence actually says:
- She said to please her and not go away: This misreads "please" as the verb "to please (someone)" rather than as a politeness word attached to a request, changing the entire meaning of the sentence.
- She told me to go away: This drops the negation completely, turning a plea to stay into an instruction to leave, the opposite of the original meaning.
- She begged me not to go away: This uses "begged", suitable for an earnest request marked by "please", keeps the negative "not to go away", and uses the natural infinitive construction expected in conversational indirect speech.
- She begged that I not go away: This keeps the correct meaning and reporting verb, but uses a subjunctive "that"-clause construction that, while grammatically defensible, is not the standard, natural transformation taught for reported requests.
The first two options change the meaning of the original sentence, and the last uses a less natural construction, leaving the direct infinitive version as the standard, expected transformation.
Therefore, the correct answer is She begged me not to go away.