Tirmidhi

Explanation:
1:
That is, if the act of fornication has been committed by him three times, then each time he will receive the punishment of flogging, and on the fourth occasion he will be exiled from the city.

2:
Because this is precisely the meaning of the hadith of the chapter: if the owner possesses reliable testimony regarding the immorality of his slave-girl or slave, then the owner himself can establish the prescribed punishment (hadd).

Hadith Commentary:

This is because such an immoral (fahishah) woman cannot remain in the house of a Muslim.

In the Noble Qur’an, Allah the Exalted has said: “Al-khabithaatu lil-khabitheena wal-khabithoona lil-khabithaat” (An-Nur: 26) — that is, impure (fornicating) women are for impure (fornicating) men, and impure (fornicating) men are for impure (fornicating) women.

Hadith Commentary:
(1)
If a slave-girl repeatedly commits fornication, then it is better to get rid of her even if it means giving her away for free.

(2)
It should be noted that the mention of carrying out the prescribed punishment (hadd) in this hadith does not mean that the owner himself should start flogging her; rather, the right to implement the hadd belongs to the court. After the hadd has been carried out, there is no need to further reproach, blame, or say anything to her, because the criminal has paid the expiation for her sin and, according to the law, has received her due punishment.

(3)
The relevance of this hadith to the chapter heading is that the command to sell a slave-girl is general in this hadith, and it includes even a mudabbara (a slave-girl set free upon the owner's death).
Allamah ‘Ayni rahimahullah has objected to this reasoning, stating that in the hadith, the permissibility of selling is conditional upon the slave-girl committing fornication two or three times, whereas according to those who permit the sale of a mudabbara, her sale is valid in all circumstances, whether she commits fornication or not.
(‘Umdat al-Qari: 8/526)
However, this objection by the aforementioned scholar is not correct, because if a mudabbara slave-girl commits fornication two or three times, the permissibility of selling her is understood from this hadith. And those who do not consider the sale of a mudabbara permissible do not accept its permissibility even in the case of fornication; therefore, this hadith is contrary to their position and in agreement with those who permit the sale of a mudabbara. Thus, the mention in the hadith is of a slave-girl in general, and this includes the mudabbara as well.
And Allah knows best.

Hadith Commentary:
(1)
Some scholars have derived from this hadith the ruling that if a slave-girl (bondwoman) commits fornication (zina), she should not be exiled, because the purpose of exile is to distance her from a corrupt environment, and this purpose is achieved by selling her. Furthermore, if she is exiled, it would become an obstacle to her sale and purchase. However, some scholars hold the view that the slave-girl should not be exiled, but after flogging, a male slave should be banished from the country for six months, because exiling a slave-girl is akin to making a woman travel without a mahram, whereas this difficulty does not arise in the case of a male slave.

(2)
In any case, the position of Imam al-Bukhari rahimahullah is that the slave-girl should be punished, but it is not correct to revile her or to exile her.
(Fath al-Bari: 203/122)
In one narration, it is mentioned that a man came to the Messenger of Allah sallallahu alayhi wa sallam and said:
“O Messenger of Allah! My slave-girl has committed fornication, and her pregnancy has made her fornication evident.”
He (the Prophet) said:
“Flog her with fifty lashes.”
(As-Sunan al-Kubra by an-Nasa’i, Hadith: 7215)
That is, the Messenger of Allah sallallahu alayhi wa sallam instructed that she be flogged with fifty lashes, but did not order her to be exiled. This also supports the position of Imam al-Bukhari rahimahullah.
And Allah knows best.

Benefits and Issues:
➊ Only the owner of the slave-girl is authorized to carry out the prescribed punishment (hadd) upon her, and it is not sufficient to merely rebuke or shame her to the extent that the prescribed legal punishment is suspended.

➋ The meaning of shaming her can also be that she should not be excessively shamed.
Because sometimes certain temperaments, when shamed in such a manner, become even more brazen, and their negative emotions are stirred up, leading them to deliberately commit sins.
This is a psychological issue.

➌ A slave or servant of corrupt nature should be kept away from oneself.