jmc

Multiplying both sides by $(x+2)(x-2)=x^2-4,$ we get $$6-x=2(x^2-4)+x(x-2).$$This simplifies to $3x^2-x-14=0.$ This factors as $(x+2)(3x-7)=0,$ so $x=-2$ or $x=\frac{7}{3}.$

Checking, we find that the given equation is not defined for $x=-2.$ Only $x=\boxed{\frac{7}{3}}$ is a solution.