Answer
This equation has no solution.
Work Step by Step
$\dfrac{4}{2x^{2}+3x-9}+\dfrac{2}{2x^{2}-x-3}=\dfrac{3}{x^{2}+4x+3}$
Factor all three rational expressions completely:
$\dfrac{4}{(x+3)(2x-3)}+\dfrac{2}{(x+1)(2x-3)}=\dfrac{3}{(x+3)(x+1)}$
Multiply the whole equation by $(x+3)(2x-3)(x+1)$:
$(x+3)(2x-3)(x+1)\Big[\dfrac{4}{(x+3)(2x-3)}+\dfrac{2}{(x+1)(2x-3)}=\dfrac{3}{(x+3)(x+1)}\Big]$
$4(x+1)+2(x+3)=3(2x-3)$
$4x+4+2x+6=6x-9$
Take all terms to the left side and simplify:
$4x+4+2x+6-6x+9=0$
$19\ne0$ False
Since simplifying resulted in a false statement, this equation has no solution.
