Hence, the process of solving polynomial inequalities involves identifying the function’s zeros and ascertaining the sign of each interval. In order to solve the inequality, it suffices to select a test-point for each interval to determine the sign of the entire interval.

Solution:

By performing some algebraic manipulations, it can be determined that the inequality to be proven is $-6asigma_1le50+15a^2$. Thus, it is necessary for $sigma_1$ to be less than or equal to $-frac {50+15a^2}{6a}$.

However, there is no maximum limit for $sigma _1$, resulting in the general inconsistency of the inequality.

When $a$ is equal to -1 and $sigma_1$ is equal to 20, the inequality does not hold.

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