baby farm
n.— «Among many of the phrases used to describe her for her despicable crime, the 33-year-old was called a “baby farm” because she appeared to deliberately increase her brood to gain more state aid.» —“Reforms hark back to roots of the welfare state” by Laura Devlin EDP24 (Norwich, England, United Kingdom) Dec. 11, 2008. (source: Double-Tongued Dictionary)
I’ve heard the word “baby factory” in this context but never baby farm.