Because He (the son of Rohini) will manifest extraordinary bodily strength, He will also be known as Bala. Moreover, because He unites two families - Vasudeva's family and the family of Nanda Maharaja - He will be known as Sankarsana - SB 10.8.12

Expressions researched:
"because He will manifest extraordinary bodily strength, He will also be known as Bala. Moreover, because He unites two families—Vasudeva's family and the family of Nanda Mahārāja—He will be known as Saṅkarṣaṇa"

Srimad-Bhagavatam

SB Canto 10.1 to 10.13

Garga Muni said: This child, the son of Rohiṇī, will give all happiness to His relatives and friends by His transcendental qualities. Therefore He will be known as Rāma. And because He will manifest extraordinary bodily strength, He will also be known as Bala. Moreover, because He unites two families—Vasudeva's family and the family of Nanda Mahārāja—He will be known as Saṅkarṣaṇa.

SB 10.8.12, Translation and Purport

Garga Muni said: This child, the son of Rohiṇī, will give all happiness to His relatives and friends by His transcendental qualities. Therefore He will be known as Rāma. And because He will manifest extraordinary bodily strength, He will also be known as Bala. Moreover, because He unites two families—Vasudeva's family and the family of Nanda Mahārāja—He will be known as Saṅkarṣaṇa.

Baladeva was actually the son of Devakī, but He was transferred from Devakī's womb to that of Rohiṇī. This fact was not disclosed. According to a statement in the Hari-vaṁśa:

pratyuvāca tato rāmaḥ

sarvāṁs tān abhitaḥ sthitān

yādaveṣv api sarveṣu

bhavanto mama vallabhāḥ

Garga Muni did disclose to Nanda Mahārāja that Balarāma would be known as Saṅkarṣaṇa because of uniting two families—the yadu-vaṁśa and the vaṁśa of Nanda Mahārāja—one of which was known as kṣatriya and the other as vaiśya. Both families had the same original forefather, the only difference being that Nanda Mahārāja was born of a vaiśya wife whereas Vasudeva was born of a kṣatriya wife. Later, Nanda Mahārāja married a vaiśya wife, and Vasudeva married a kṣatriya wife. So although the families of Nanda Mahārāja and Vasudeva both came from the same father, they were divided as kṣatriya and vaiśya. Now Baladeva united them, and therefore He was known as Saṅkarṣaṇa.