The roots of polynomial $ p(s) $ are:
$$ \begin{aligned}s_1 &= -0.4384\\[1 em]s_2 &= -4.5616 \end{aligned} $$The solutions of $ s^2+5s+2 = 0 $ are: $ s = -\dfrac{ 5 }{ 2 }-\dfrac{\sqrt{ 17 }}{ 2 } ~ \text{and} ~ s = -\dfrac{ 5 }{ 2 }+\dfrac{\sqrt{ 17 }}{ 2 }$.
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