Former Massachusetts Gov. Deval Patrick has joined Boston private-equity firm Bain Capital LLC to lead a new unit that will focus on investments aiming for financial returns as well as measurable social good.

Mr. Patrick, a Democrat and ally of President Barack Obama, left the Massachusetts statehouse in January after serving two four-year terms. In joining Bain, Mr. Patrick is taking a job at a firm co-founded by Mitt Romney, the unsuccessful Republican presidential nominee who preceded Mr. Patrick in the Massachusetts governor’s...