#### Answer

The solution is $x=-9$

#### Work Step by Step

$\dfrac{4}{x^{2}+x-6}-\dfrac{1}{x^{2}-4}=\dfrac{2}{x^{2}+5x+6}$
Factor all rational expressions completely:
$\dfrac{4}{(x+3)(x-2)}-\dfrac{1}{(x-2)(x+2)}=\dfrac{2}{(x+3)(x+2)}$
Multiply the whole equation by $(x+3)(x-2)(x+2)$:
$(x+3)(x-2)(x+2)\Big[\dfrac{4}{(x+3)(x-2)}-\dfrac{1}{(x-2)(x+2)}=\dfrac{2}{(x+3)(x+2)}\Big]$
$4(x+2)-(x+3)=2(x-2)$
Evaluate the indicated operations:
$4x+8-x-3=2x-4$
Take all terms with $x$ to the left side and all terms without $x$ to the right side:
$4x-x-2x=-4-8+3$
Simplify both sides:
$x=-9$