$f(x)=\dfrac{-12 e^x - 20}{e^{2x}+6e^x+5}$

let $u=e^x$

$f(u)=\dfrac{-12u-20}{u^2+6u+5}$

$f(u)=\dfrac{-12u-20}{(u+5)(u+1)}$

$f(u) = \dfrac A {u+5} + \dfrac B {u+1}$

$A(u+1) + B(u+5) = -12u - 20$

You should be able to finish this from here. Just substitute back $e^x$ for $u$ when you are finished.