Answer
$x=9$
Work Step by Step
$\frac{x-6}{x+3}+\frac{2x}{x-3}=\frac{4x+3}{x+3}$
$\frac{x-6}{x+3}*(x-3)/(x-3)+\frac{2x}{x-3}*(x+3)/(x+3)=\frac{4x+3}{x+3}*(x-3)/(x-3)$
$\frac{(x-6)(x-3)}{(x+3)(x-3)}+\frac{(2x)(x+3)}{(x-3)(x+3)}=\frac{(4x+3)(x-3)}{(x+3)(x-3)}$
$((x-6)(x-3)+(2x)(x+3))/{(x-3)(x+3)}=\frac{(4x+3)(x-3)}{(x+3)(x-3)}$
$((x-6)(x-3)+(2x)(x+3))=(4x+3)*(x-3))$
$x^2-9x+18 +2x^2+6x = 4x^2-12x+3x-9$
$3x^2-3x+18 = 4x^2-9x-9$
$3x^2-3x+18-3x^2+3x-18 = 4x^2-9x-9-3x^2+3x-18$
$0 = x^2-6x-27$
$(x-9)(x+6)=0$
$x-9=0$
$x=9$
$x+6=0$
$x=-6$
$x=9$
$\frac{x-6}{x+3}+\frac{2x}{x-3}=\frac{4x+3}{x+3}$
$\frac{9-6}{9+3}+\frac{2*9}{9-3}=\frac{4*9+3}{9+3}$
$\frac{3}{12}+\frac{18}{6}=\frac{39}{12}$
$.25 + 3 = 3.25$
$x=-6$
$\frac{x-6}{x+3}+\frac{2x}{x-3}=\frac{4x+3}{x+3}$
$\frac{-6-6}{-6+3}+\frac{2*-6}{-6-3}=\frac{4*-6+3}{-6+3}$
$\frac{-12}{-3}+\frac{-12}{-9}=\frac{-21}{-9}$
$4 +4/3 = 7/3$ (not true, so this answer is invalid)
