Answer
This equation has no solution.
Work Step by Step
$\dfrac{2}{x^{2}-2x-3}+\dfrac{5}{x^{2}-x-6}=\dfrac{1}{x^{2}+3x+2}$
Factor all three rational expressions completely:
$\dfrac{2}{(x-3)(x+1)}+\dfrac{5}{(x-3)(x+2)}=\dfrac{1}{(x+2)(x+1)}$
Multiply the whole equation by $(x-3)(x+1)(x+2)$:
$(x-3)(x+1)(x+2)\Big[\dfrac{2}{(x-3)(x+1)}+\dfrac{5}{(x-3)(x+2)}=\dfrac{1}{(x+2)(x+1)}\Big]$
$2(x+2)+5(x+1)=x-3$
$2x+4+5x+5=x-3$
Take all terms to the left side and simplify:
$2x+4+5x+5-x+3=0$
$6x+12=0$
Take $12$ to the right side:
$6x=-12$
Take $6$ to divide the right side:
$x=-\dfrac{12}{6}$
$x=-2$
The original equation is undefined for $x=-2$, so this equation has no solution.
