To carry out, and to do before an audience. Both mean acting in a way that can be judged.
Old French parfournir (to accomplish fully), from par- (thoroughly) + fournir (to furnish, provide); to carry a thing all the way through, whether the audience is a task or a crowd.
To carry a thing out, or to do it where others can watch — either way, an act done in a way that can afterward be judged.