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- Feb 14, 2012

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I have encountered a problem recently for which I couldn't think of even a single method to attempt it, and this usually is an indicator that a problem really isn't up my alley. That notwithstanding, I don't wish yet to concede defeat. Could someone please show me at least some idea on how to crack it? Thanks in advance.

Problem:

Show that the five roots of the quintic $a_5x^5+a_4x^4+a_3x^3+a_2x^2+a_1x+a_0=0$ are not all real if $2a_4^2<5a_5a_3$.