The people considered part of the labor force are those who are either employed or actively seeking employment. Based on the options given:
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A 14-year-old with an after-school job - This person is employed, but in many places, there are restrictions on the age at which one can formally enter the labor force.
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An army sergeant - This person is employed and therefore is part of the labor force.
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A person who has lost a job but is looking for one - This person is actively seeking employment and is thus part of the labor force.
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A mother raising her children at home - Unless she is actively seeking employment, she is not considered part of the labor force.
Based on this analysis, the people who are clearly part of the labor force are the army sergeant and the person who has lost a job but is looking for one.