Answer
$9$.
Work Step by Step
The given expression is
$\Rightarrow \frac{x-6}{x+3}+\frac{2x}{x-3}=\frac{4x+3}{x+3}$
Multiply the equation by $(x+3)(x-3)$.
$\Rightarrow (x+3)(x-3)\left ( \frac{x-6}{x+3}+\frac{2x}{x-3} \right )=(x+3)(x-3)\left (\frac{4x+3}{x+3}\right )$
Use the distributive property and divide out common factors.
$\Rightarrow (x-3)(x-6)+2x( x+3 )=(x-3)(4x+3)$
Simplify.
$\Rightarrow x^2-9x+18+2x^2+6x=4x^2-9x-9$
Move all terms to the one side.
$\Rightarrow x^2-9x+18+2x^2+6x-4x^2+9x+9=0$
Add like terms.
$\Rightarrow -x^2+6x+27=0$
Multiply by $-1$.
$\Rightarrow x^2-6x-27=0$
Factor.
$\Rightarrow (x-9)(x-3)=0$
Use zero product property.
$x-9=0$ or $x-3=0$
Solve for $x$.
$x=9$ or $x=3$
Check $x=9$.
$\Rightarrow \frac{9-6}{9+3}+\frac{2(9)}{9-3}=\frac{4(9)+3}{9+3}$
$\Rightarrow \frac{3}{12}+\frac{18}{6}=\frac{39}{12}$
$\Rightarrow \frac{1}{4}+\frac{3}{1}=\frac{13}{4}$
$\Rightarrow \frac{1+12}{4}=\frac{13}{4}$
$\Rightarrow \frac{13}{4}=\frac{13}{4}$
Check $x=3$.
$\Rightarrow \frac{3-6}{3+3}+\frac{2(3)}{3-3}=\frac{4(3)+3}{3+3}$
$\Rightarrow \frac{-3}{6}+\frac{6}{0}=\frac{15}{6}$
Undefined.
Hence, the correct solution is $x=9$
