The Bahmani Sultanate emerged as a powerful kingdom in the Deccan after breaking away from the:

Delhi Sultanate
Vijayanagara Empire
Mughal Empire
Maratha Confederacy

The correct answer is: A) Delhi Sultanate.

The Bahmani Sultanate was a Muslim sultanate that ruled over the Deccan Plateau in India from 1347 to 1518. It was founded by Alauddin Bahman Shah, a former governor of the Delhi Sultanate, after he broke away from the Delhi Sultanate in 1347. The Bahmani Sultanate was a powerful kingdom that reached its peak under the rule of Muhammad Shah I (1358-1375). The Bahmani Sultanate was eventually weakened by internal conflicts and was conquered by the Mughal Empire in 1518.

The Vijayanagara Empire was a Hindu empire that ruled over the Deccan Plateau in India from the 14th to the 16th centuries. It was founded by Harihara I and Bukka Raya I, two brothers who were feudatories of the Hoysala Empire. The Vijayanagara Empire reached its peak under the rule of Krishnadeva Raya (1509-1529). The Vijayanagara Empire was eventually weakened by internal conflicts and was conquered by the Deccan sultanates in the 16th century.

The Mughal Empire was a Muslim empire that ruled over most of the Indian subcontinent from the 16th to the 18th centuries. It was founded by Babur, a descendant of Timur and Genghis Khan. The Mughal Empire reached its peak under the rule of Akbar the Great (1556-1605). The Mughal Empire was eventually weakened by internal conflicts and was overthrown by the British East India Company in the 18th century.

The Maratha Confederacy was a Hindu confederacy that ruled over much of the Indian subcontinent in the 17th and 18th centuries. It was founded by Shivaji Maharaj, who declared independence from the Mughal Empire in 1674. The Maratha Confederacy reached its peak under the rule of Bajirao I (1720-1740). The Maratha Confederacy was eventually weakened by internal conflicts and was conquered by the British East India Company in the 19th century.