The equation \(m^2 + 380 = 381m\) does not satisfy m=-20, the equation \(m^3 + 380 = 381m\) does.

\(m^3 + 380 = 381m\) or

\(m^3 + 380 = 380m + m\) or

rearranging

\(m^3 - m = 380m -380\)

\(m(m^2 - 1) = 380(m -1)\)

now \((m^2 - 1)\) can be rewritten as \((m+1)(m-1)\).

\(m(m + 1)(m - 1) = 380(m -1)\) since m is a negative integer we can cancel out m-1 terms from both sides.

so \(m(m+1)=380\) .

Clearly m = -20 the only possible answer.
_________________

Sandy
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