The word "اسْتَمْتَعْتُم"
The phrase those whom you profit by, in Arabic istamtaˤtum bihi, has the same root as Mut‘a, that is, MTĦ. The word is usually translated in its general meaning as "those whom ye seek content" or "they whom you have enjoyed". If the word was translated as an Islamic term, the translation would be: "They with whom you have made Mut'a", or "They whom you have married for a fixed time".
Muslims differ on what is meant by Mut'a here, and which judgment the verse gives about it. Generally, Shi'i Muslims tend to believe that Mut'a here refers to the temporary marriage, and that this verse permits it.
Among the Sunnis, different view exists. Some Sunnis do not believe that this verse refers to the Nikah-e-Mut'a at all. This view was favored by Suyuti. Other Sunnis agree that the verse refers to the temporary marriage, but disagree that the verse permits it. This view was favored by Ibn Kathir.
According to Sachiko Murata, a scholar of Theology from Tehran University:
(“Temporary Marriage in Islamic Law” Published in Al-Serat, Volume, 13, Issue 1, in 1979, available online at: http://www.al-islam.org/al-serat/muta/5.htm)
There are many sayings have been related from the Companions of the Prophet (S) and those who followed them (al-taibi'un) confirming the Shi'i view that verse 24 of this chapter concerns mut 'a. Several of the companions, including Ibn 'Abbas, the ancestor of the 'Abbasid caliphs, Ibn Mas'ud, one of the first to accept Islam, and Ubayy b. Ka'b, one of the scribes of the revelation, hold that three words have been dropped form this passage in the Qur'an and that the original version read: 'So those of them whom you enjoy to a specified term (ila ajal musamma).' Although Shi'is do not believe that Qur'an that we read today has altered,this clearly indicates the view of these renowned salaf that the verse refers to mut'a. (See “Temporary Marriage in Islamic Law” Published in Al-Serat, Volume, 13, Issue 1, in 1979, available online at: http://www.al-islam.org/al-serat/muta/5.htm)
Tabari continues to say that Ubay Ibn Ka'ab also knew that tafsir.
The next part of this same verse states as follows: 'So those of them whom you enjoy, give them their appointed wages.' The word 'so' (fa) shows that this part of the verse is the conclusion reached by the previous words. This section is either part of the previous subject matter, or an example of it; in other words, its relation to the previous section is either that of the part which is completing the whole, or the particular example to the universal principle. And since the previous section deals with the different kinds of legitimate sexual relationships, either by marriage or the purchase of slaves, we can conclude that this section of the verse is the exposition of a further kind of marriage, not mentioned previously; a kind which requires that the man pay the wages of his wife.
In Majma' al-bayan, al Tabarsi, the famous Shi'i commentator of the Qur'an summarizes the Shi'i arguments: the word 'enjoy' in this verse refers to the marriage of mut'a, i.e., a marriage for a specified dower and a determined time period. This opinion has been related from Ibn 'Abbas and many of the 'followers' of the Companions such as Isma'il b. 'Abd al-Rahman al-Suddi (d. 127/744-45) and Sa'id b. Jubayr al-Asadi (95/713-14). In fact, this clearly must be the case, for although the words istimta' and mut'a have the literal meaning of 'enjoyment', in the language of the shari'a they refer to the contract of temporary marriage, especially when they are followed by the word 'women'. Hence the meaning of the verse is: 'Whenever you draw up a contract of mut'a with a woman, you must pay her, her wages.'
The phrase "give them their dowries as appointed" is "bihi minhunna fātūhunna ujūrahunna farīzatan". "'Ujūrahunna" is derived from ajr "compensation/payment". This is in contrast to the verse on the subject of Nikah (Nikah e Daaimi), verse 4:4. It states:
The section "and there is no blame on you about what you mutually agree after what is appointed" refers to prolonging the marriage or making it permanent, something which is mustahab "recommended". If this doesn't occur, the wife must observe the Iddah, making it impossible for a woman to engage in a Nikah-e-Mut'a, and have sexual intercourse with more than six partners per year, assuming a menstruation cycle of 20 days.
Tabari in his Tafsir al-Kabir under the verse 4:24 mentioned a Tafsir Qudsi, a narration attributed to Abu Nadra says:
(“Temporary Marriage in Islamic Law” Published in Al-Serat, Volume, 13, Issue 1, in 1979, available online at: http://www.al-islam.org/al-serat/muta/5.htm)