#### Answer

The equation has no solution.

#### Work Step by Step

$\dfrac{1}{x-4}-\dfrac{5}{x+2}=\dfrac{6}{x^{2}-2x-8}$
Factor the denominator of the fraction on the right side:
$\dfrac{1}{x-4}-\dfrac{5}{x+2}=\dfrac{6}{(x+2)(x-4)}$
Multiply the whole equation by $(x+2)(x-4)$:
$(x+2)(x-4)\Big[\dfrac{1}{x-4}-\dfrac{5}{x+2}=\dfrac{6}{(x+2)(x-4)}\Big]$
$(x+2)-5(x-4)=6$
$x+2-5x+20=6$
Take all terms without $x$ to the right side of the equation, simplify, and solve for $x$:
$x-5x=6-2-20$
$-4x=-16$
$x=\dfrac{-16}{-4}$
$x=4$
Since the original equation is undefined for $x=4$, it has no solution.