Long but lovely equation

We have, $\dfrac{(x+1)(x-3)}{5(x+2)(x-4)}+\dfrac{(x+3)(x-5)}{9(x+4)(x-6)}-\dfrac{2(x+5)(x-7)}{13(x+6)(x-8)}=\dfrac{92}{585}.$

Observing the numerators and denominators,we conclude that each of them contain the expression $x^2-2x$ ,so put $\color{#3D99F6}{(x-1)^2}=\color{#3D99F6}{y}$ ,our equation becomes $\dfrac{\color{#3D99F6}{y}-4}{5(\color{#3D99F6}{y}-9)}+\dfrac{\color{#3D99F6}{y}-16}{9(\color{#3D99F6}{y}-25)}-\dfrac{2(\color{#3D99F6}{y}-36)}{13(\color{#3D99F6}{y}-49)}=\dfrac{92}{585}.$ Now note that $\dfrac{1}{5}+\dfrac{1}{9}-\dfrac{2}{13}=\dfrac{92}{585}$ .Subtracting corresponding terms we get,

$\dfrac{5}{5(\color{#3D99F6}{y}-9)}+\dfrac{9}{9(\color{#3D99F6}{y}-25)}-\dfrac{2\cdot13}{13(\color{#3D99F6}{y}-49)}=0 \implies \dfrac{1}{\color{#3D99F6}{y}-9}+\dfrac{1}{\color{#3D99F6}{y}-25}-\dfrac{2}{\color{#3D99F6}{y}-49}=0 \\ \implies \dfrac{1}{\color{#3D99F6}{y}-9}-\dfrac{1}{\color{#3D99F6}{y}-49}=\dfrac{1}{\color{#3D99F6}{y}-49}-\dfrac{1}{\color{#3D99F6}{y}-25} \implies \dfrac{-40}{\color{#3D99F6}{y}-9}=\dfrac{24}{\color{#3D99F6}{y}-25} \\ \implies 3(\color{#3D99F6}{y}-9)+5(\color{#3D99F6}{y}-25)=0 \implies 8\color{#3D99F6}{y}=152 \\ \therefore \color{#3D99F6}{y}=19 \\ \therefore ~\boxed{x=1\pm\sqrt{19}}$

$Let~~ t=x^2-2x,~~ and~~ n=8.~~~ \text{Then the equation after multiplying the factors is:-} \\ \dfrac{t - 3}{5*(t - n)} + \dfrac{t - 15}{9*(t - 3n)} - 2*\dfrac{t - 35}{13*(t - 6n)} =\dfrac{92}{585}.\\ \implies~ \dfrac{(t - n) + 5}{5*(t - n)} + \dfrac{(t - 3n) + 9}{9*(t - 3n)} - 2*\dfrac{(t - 6n) + 13}{13*(t - 6n)} =\dfrac{92}{585}.\\ \implies~\dfrac 1 5 + \dfrac 1{t - n} + \dfrac 1 9 + \dfrac 1{t - 3n} - 2*\dfrac 1 5 - 2* \dfrac 1{t -6n} - \dfrac{92}{585}=0.\\ But ~\dfrac 1 5+\dfrac 1 9 - \dfrac 2{13} -\dfrac{92}{585}=0.\\ \therefore~~ \dfrac 1{t - n} + \dfrac 1{t -3n} - \dfrac 2{t - 6n}=0.\\ \implies~ \dfrac{(t - 3n)+(t - n)}{(t - n)*(t - 3n)} - \dfrac 2{t - 6n}=0\\ But ~~ only ~~ numerator = 0, ~with ~common ~ denominator.\\ \implies~ (2t - 4n)*(t - 6n) - 2*(t - n)*(t - 3n)=0\\ \implies~ - 16nt +24n^2 + 8nt - 6n^2=0, ~~n\neq 0.\\ \therefore ~ -8t + 18n=0, ~~putting ~~ values~~ of~~ t~~ and~~ n,\\ We ~ get~~x^2 -2x - 18=0, \therefore ~x= 1 \pm \sqrt{19}=a+\sqrt b.\\ a+b=\Large ~~~~\color{#D61F06}{20}$