Vote out refers to a collective decision-making process, usually in a democratic context. The majority voted out the policy-maker due to dissatisfaction with his performance.
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Negative Connotation
The phrase often implies that a person is being removed for negative reasons, such as failure or misconduct. After the scandal, the board voted out the CEO.
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Election Context
Using vote out usually means it's related to an election or formal vote by a group or committee. Activists campaigned to vote out politicians who opposed environmental reforms.