。
The adopted child lived happily with the new family.
Literal
Adopted-child [into-に] became [attributive-た] child [topic-は] new family [with-と] happy life [object-を] did.
養子 ('adopted child') — 養 ('raise / nurture') + 子 ('child'). 養子になる ('become an adopted child' = 'be adopted'). 幸せな生活をする ('live a happy life') is a standard collocation where 生活 ('life / living') takes する as its verb and a modifier as adjective.