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You cannot prove it because it is not true. Some quintics ARE solvable.

For example, if all the coefficients (in a quintic in x) sum to 0 then (x - 1) is a factor. So one solution is x = 1 and you are left qith a quartic. If the sum of odd coeffs equals the sum of even coeffs then (x + 1) is a factor. So, in some cases, at least, quintics are solvable.

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Q: How do you prove the quintic is not solvable?
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