Given,

$6x^6-25x^5+31x^4-31x^2+25x-6=0$

Let $a_1,a_2, a_3, a_4, a_5, a_6$ be roots of the given equation,

Now given that sum of two roots is $\frac{5}{3}$,

And by hit and trial we get other two roots as $1 \& -1$,

So, assuming $a_1+a_2=\frac{5}{2} \& a_3=1, a_4=-1$

By formula of sum of roots $=\frac{-\left(-25\right)}{6}$ we get,

$a_1+a_2+a_3+a_4+a_5+a_6=\frac{25}{6}$

$\Rightarrow \frac{5}{2}+1+\left(-1\right)+a_5+a_6=\frac{25}{6}$

$\Rightarrow a_5+a_6=\frac{25}{6}-\frac{5}{2}$

$\Rightarrow a_5+a_6=\frac{5}{3}$