If  $\left(\alpha , {\alpha }^2\right)$ lies inside the triangle formed by the lines $2x+3y-1=0$,  $x+2y-3=0$,$5x-6y-1=0$  then

O and the point $(\alpha ,{\alpha }^2)$lie on the opposite  sides

With respect to $2x+3y-1=0$

  $\Rightarrow \boxed{2\alpha +3{\alpha }^2-1>0}$

   O and the point $(\alpha ,{\alpha }^2)$ lie on the same side with respect to $x+2y-3=0$

 $\Rightarrow \boxed{\alpha +2{\alpha }^2-3<0}$

Again O and the point $(\alpha ,{\alpha }^2)$lie on the same side w. r. t

$\Rightarrow 5\alpha -6{\alpha }^2-1<0$

$\Rightarrow \boxed{6{\alpha }^2-5\alpha +1>0}$