Assalamualaikum ustaz. Is it permissible to perform the Dhuha prayer in four rakaat and one salam? Or it must be performed in two rakaats one salam and continue with another two rakaats and one salam? Hope for an explanation.
Waalaikumussalam wrt. wbt.,
Alhamdulillah, praise and gratitude to Allah SWT for His countless blessings for us all. Praise and salutations to our beloved Prophet Muhammad PBUH, his family, companions, and all those who follow his footsteps until the Final day.
Dhuha prayer is one of the encouraged nafilah (sunnah) prayers. There are numerous evidences which state the advantages and importance of this prayer. Among them is stated by Abu Hurairah RA, he said:
أَوْصَانِي خَلِيلِي صلى الله عليه وسلم بِثَلاَثٍ بِصِيَامِ ثَلاَثَةِ أَيَّامٍ مِنْ كُلِّ شَهْرٍ وَرَكْعَتَىِ الضُّحَى وَأَنْ أُوتِرَ قَبْلَ أَنْ أَرْقُدَ
My friend (the Holy Prophet, may peace be upon him) has instructed me to do three things: Three fasts during every month, two rak’ahs of the forenoon prayer, and observing Witr prayer before going to bed.” 
The above hadith clearly shows that the Dhuha prayer is instructed by Rasullullah PBUH. Two rakaats dhuha prayer is mentioned in the hadith explaining the briefest number of rakaat for dhuha prayer. However, it isn’t limited to only two rakaat and the jurists of Syafieyyah scholars explained that the most prioritised number of rakaats for dhuha prayer is 8 while its maximum number of rakaats is 12 rakaats as stated by Imam al-Nawawi.
Imam al-Nawawi also said: Indeed, the Prophet PBUH performed this dhuha prayer at certain times and leave it at other times, fearing that some might assume that it is obligatory or that it might be made obligatory for them. 
Coming back to the above question, it is sunnah to perform the dhuha prayer in two rakaats and one salam, but it is also permissible to perform it in four rakaats and one salam. 
However, in our opinion, the dhuha prayer should be performed in two rakaats and one salam for a total of four rakaats or more because this is closer to the evidences. From Ummu Hani R.Anha, she said:
أَنَّ رَسُوْلَ اللهِ صَلَّى يَوْمَ الْفَتْحِ سُبْحَةَ الضُّحَى ثَمَانِيَ رَكَعَاتٍ وَيُسَلِّمُ مِنْ كُلِّ رَكْعَتَيْنِ
“On the day of the Conquest (of Makkah) the Messenger of Allah (ﷺ) prayed voluntary Duha with eight Rak’ah, saying the Salam after each two Rak’ah.” 
Furthermore, in a hadith narrated from Abdullah bin Umar R.Anhuma, Rasullullah PBUH said:
صَلاَةُ اللَّيْلِ وَالنَّهَارِ مَثْنَى مَثْنَى
“(Voluntary) prayer in both the day and night is two at a time.” 
This sahih hadith explained that it is afdhal (prioritized) to perform the sunnah prayers in two rakaats and one salam whether the sunnah prayer is performed during the day or at night. Thus, it is encouraged to perform the sunnah prayer in two rakaats and one salam. However, combining the rakaats with one salam is permissible. 
The opinion which states that sunnah prayers are divided to two rakaats and one salam is the opinion of the majority of scholars including the scholars of madhhab Syafi’e. 
 Narrated by Muslim (721)
 See al-Majmu’ Syarah al-Muhazzab, 4/37
 See I`anah al-Talibin, 1/294
 Narrated by Abu Daud (1290). Imam al-Nawawi said: Our ashab kami and jumhur of scholars uses this hadith to support their statement in saying the istihbab (encouragement)of performing eight rakaats of dhuha prayer. (See al-Minhaj Syarh Sahih Muslim, 5/233)
 Narrated by Abu Daud (1295)
 See Syarh al-Nawawi ‘ala Sahih Muslim, 6/30
 See Tuhfah al-Ahwazi, 2/416, 2/411