The correct answer is Raj Darbhanga.
Raj Darbhanga was a princely state in the Indian subcontinent, which existed from the 16th century to 1947. It was located in the present-day Indian state of Bihar, and controlled a vast territory in the present-day districts of Gopalganj, Siwan, and Saran.
The state was founded by Raja Ram Singh in the 16th century. He was a descendant of the Suryavanshi Rajputs, and his ancestors had ruled over the region for centuries. Raja Ram Singh was a powerful and influential ruler, and he expanded his territory significantly during his reign. He also built a number of temples and palaces in the state, and he is considered to be one of the most important rulers in the history of Bihar.
The state of Darbhanga continued to exist until 1947, when it was merged with the Indian Union. However, the family of the former rulers of Darbhanga still maintains a significant presence in the state, and they are considered to be one of the most powerful and influential families in Bihar.
The other options are incorrect. Tekari Raj was a princely state in the Indian subcontinent, which existed from the 17th century to 1947. It was located in the present-day Indian state of Madhya Pradesh, and controlled a vast territory in the present-day districts of Chhatarpur, Panna, and Damoh. Hathwa Raj was a princely state in the Indian subcontinent, which existed from the 16th century to 1947. It was located in the present-day Indian state of Uttar Pradesh, and controlled a vast territory in the present-day districts of Basti, Siddharthnagar, and Maharajganj. Banaili Raj was a princely state in the Indian subcontinent, which existed from the 17th century to 1947. It was located in the present-day Indian state of Bihar, and controlled a vast territory in the present-day districts of Begusarai, Khagaria, and Lakhisarai.