Write A for alpha and B for beta.
(x-A)(x-B)=x^2-8x+6 so A+B=8 and AB=6.
Polynomial with zeroes 1/A^2 and 1/B^2 is (x-1/A^2)(x-1/B^2)=x^2-x(1/A^2+1/B^2)+1/A^2B^2.
1/A^2B^2=1/(AB)^2=1/36.
1/A^2+1/B^2=(B^2+A^2)/(AB)^2=((B^2+2AB+A^2)-2AB)/36=
((A+B)^2-12)/36=(64-12)/36=52/36=13/9.
The polynomial is x^2-13x/9+1/36 or 36x^2-52x+1.