게 풀어내다

unpack and present it well

풀어내다 is taking something knotty and working it all the way out into finished form — solving it, or laying it out so an audience can follow. The -게 adverb says in what style, and that is where the meaning lives: [일상 속 궁금증을 쉽고 재미있게 풀어내는 지식 예능], 유쾌하게 풀어냈습니다, 아름답게 풀어낸 발라드록 곡. It also covers plain problem-solving (이런 걸 수학적으로 어떻게 풀어내는 거지?). Several lines here read like show or album descriptions rather than casual talk.

Register: descriptive, often promotional