Question 1059124
I will do only your first example, and assume your groupings are exactly as expressed.


8/x+9=5/2x+(15/4)


{{{8/x+9=(5/2)x+(15/4)}}}


{{{4x(8/x+9)=4x((5/2)x+(15/4))}}}


{{{32+36x=10x^2+15x}}}


{{{10x^2+15x-36x-32=0}}}


{{{10x^2-21x-32=0}}}-------easier to use general formula solution than to try checking for factorization.


{{{x=(21+- sqrt(21^2-4*10*(-32)))/(2*10)}}}


{{{x=(-21+- sqrt(441+320))/(2*10)}}}


{{{highlight(x=(-21+- sqrt(761))/(20))}}}, and the discriminant might but probably has no useful factorizations.