The region known for its tradition of terracotta work in Bihar is:

Darbhanga
Patna
Gaya
Bhagalpur

The correct answer is: Bhagalpur.

Bhagalpur is a city in the Indian state of Bihar. It is the headquarters of Bhagalpur district and Bhagalpur division. The city is located on the banks of the Ganges River, about 200 kilometers (120 mi) north of Patna, the state capital.

Bhagalpur is known for its terracotta work. Terracotta is a type of unglazed pottery made from clay. It is often used to make decorative items, such as sculptures, pots, and tiles. The terracotta work of Bhagalpur is known for its intricate designs and bright colors.

The terracotta work of Bhagalpur has a long history. It is believed that the tradition of terracotta work in Bhagalpur began in the 16th century. The terracotta work of Bhagalpur was influenced by the Mughal style of art. The Mughals were a Muslim dynasty that ruled over India from the 16th to the 18th centuries.

The terracotta work of Bhagalpur is made by hand. The clay is first kneaded and then shaped into the desired form. The shapes are then dried and fired in a kiln. The terracotta work of Bhagalpur is often decorated with intricate designs. The designs are made by using a variety of tools, such as brushes, knives, and chisels.

The terracotta work of Bhagalpur is used for a variety of purposes. It is used to make decorative items, such as sculptures, pots, and tiles. It is also used to make functional items, such as water pots and cooking utensils.

The terracotta work of Bhagalpur is a popular tourist attraction. There are a number of shops in Bhagalpur that sell terracotta products. The terracotta work of Bhagalpur is also exported to other countries.

The other options are incorrect because they are not known for their tradition of terracotta work. Darbhanga is a city in the Indian state of Bihar. It is the headquarters of Darbhanga district and Darbhanga division. The city is located on the banks of the Kosi River, about 200 kilometers (120 mi) north of Patna, the state capital.

Patna is the capital of the Indian state of Bihar. It is the third-largest city in Bihar, after Patna and Gaya. The city is located on the banks of the Ganges River, about 200 kilometers (120 mi) north of Kolkata, the capital of West Bengal.

Gaya is a city in the Indian state of Bihar. It is the headquarters of Gaya district and Gaya division. The city is located on the banks of the Phalgu River, about 100 kilometers (60 mi) south of Patna, the state capital.

The terracotta work of Darbhanga, Patna, and Gaya is not as well-known as the terracotta work of Bhagalpur.