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A value c for which the given equation has 2 distinct negative roots.

$ x^4 + 2cx^2 + 2cx + 1 + x^2 = 0$

I solved it up to here, what does this imply ?

$ \sqrt{ 1+c^2 } > 2 - c $

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    Could you should how you arrived at this inequality? It's not correct.2012-12-16

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If we square both sides, we get $1+c^2>(2-c)^2=4-4c+c^2$, which after cancelling $c^2$ and rearranging tells us that

$ c>\frac{3}{4} $

This is assuming your work up to this point is ok, which I'm not sure it is - if $c=1$, your equation has no real roots.