$\dfrac{5x-3}{x+2}-4=\dfrac{6}{x+2}$
$↔\dfrac{5x-3}{x+2}=\dfrac{6}{x+2}+4$
$↔\dfrac{5x-3}{x+2}=\dfrac{14+4x}{x+2}$
$↔\dfrac{5x-3}{x+2}-\dfrac{14+4x}{x+2}=0$
$↔\dfrac{5x-3-(14+4x)}{x+2}=0$
$↔\dfrac{x-17}{x+2}=0$
$↔\dfrac{x}{x+2}-\dfrac{17}{x+2}=0$
$↔\dfrac{x}{x+2}=\dfrac{17}{x+2}$
$↔x(x+2)=17(x+2)$
$↔x=17$