KNOW THAT TO seek protection from Shayṭān after the opening du‘ā’ is Sunnah by agreement [of Shāfi‘ī scholars]. It is a prelude to recitation. Allah 
says: “And when you recite the Qur’ān, seek protection from Shayṭān the rejected.” [1].
says: “And when you recite the Qur’ān, seek protection from Shayṭān the rejected.” [1].
Fa-idhā qara’ta-l-Qur’āna fa-sta‘idh bi-llāhi mina-sh-shayṭāni-r-rajīm.
The meaning of this, according to the majority of the ‘ulamā’, is: “If you intend to recite the Qur’ān, seek the protection. . ..” The preferred words for seeking protection are: I seek protection in Allah from Shayṭān the rejected. One may also say: I seek protection in Allah, the All-Hearing and the All-Knowing, from Shayṭān the rejected; there is no harm in this, but the better-known and preferred of these formulae is the first.
A‘ūdhu bi-llāhi mina-sh-shayṭāni-r-rajīm. A‘ūdhu bi-llāhi-s-Samī‘i-l-‘Alīmi mina-sh-shayṭāni-r-rajīm.
We relate in the Sunans of Abū Dāwūd, at-Tirmidhī, an-Nasā’ī, Ibn Mājah, al-Bayhaqī and others that the Messenger of Allah 
said before his recitation in ṣalāh: “I seek protection in Allah, from Shayṭān the rejected; from his conceit, his [obscene] poetry and his slander.” [2].Another narration has: “I seek refuge in Allah the All-Hearing and All-knowing from the accursed Shayṭān, his [obscene] poetry, his pride and his madness.”
said before his recitation in ṣalāh: “I seek protection in Allah, from Shayṭān the rejected; from his conceit, his [obscene] poetry and his slander.” [2].Another narration has: “I seek refuge in Allah the All-Hearing and All-knowing from the accursed Shayṭān, his [obscene] poetry, his pride and his madness.”
A‘ūdhu bi-llāhi-s-Samī‘i-l-‘Alīmi mina-sh-shayṭāni-r-rajīm, min nafkhihi wa nafathihi wa hamzih.
The commentary on this verse, as stated, has been narrated in the hadith that ‘hamzuhu’ means ‘his madness’, ‘nafkhuhu’ means ‘his arrogance’ and ‘nafatḥuhu’ means ‘his poetry’ and Allah knows best.
Know that seeking protection (ta‘awwudh) is mustaḥabb and not compulsory. If one leaves it out he is not sinning and his ṣalāh is not invalid. If one does not say it in the first rak‘ah, he should say it in the second, and if one does not say it then, he should do so after that. If one seeks protection in the first, does he have to say it in the second? There are two opinions. The more correct view is that it is mustaḥabb but is more emphasised in the first. If one seeks protection in those ṣalawāt which are silent, he should do so silently. If one does so in those in which the recitation is aloud, should he say it aloud? There is disagreement on this. Some of our [Shāfi‘ī] colleagues say that one should say it softly. The majority say that ash-Shāfi‘ī had two rulings on the issue. The first is that saying it aloud and silently are equal, and that is from the text that is in al-Umm. The second is that saying it aloud is the Sunnah, and that is from text of al-Imlā’. Some of them say that the two rulings are: first, out loud, and second, silently.
The sound position overall is that one should say it out loud, and Shaykh Abū Ḥāmid al-Isfarā’īnī, the imam of our Iraqi colleagues and the companion of al-Maḥāmilī and others, corroborated this. That is what Abū Hurayrah 
used to do, and Ibn ‘Umar 
used to do it silently, which is the more correct position acording to the majority of our colleagues and is the chosen position; and Allah knows best.
used to do, and Ibn ‘Umar used to do it silently, which is the more correct position acording to the majority of our colleagues and is the chosen position; and Allah knows best.References
[1] Qur’ān 16:98.
[2] Abū Dāwūd (764), at-Tirmidhī (242), al-Bayhaqī, and an-Nasā’ī in al-Kubrā. Ibn Ḥajar grades it as ḥasan and mentions narrations which support it.