[tex]x^2-x-12=x^2+3x-4x-12=x(x+3)-4(x+3)=(x+3)(x-4)\\\\x^2-2x-8=x^2+2x-4x-8=x(x+2)-4(x+2)=(x+2)(x-4)\\\\4x^2=2x\cdot2x[/tex]
[tex] \dfrac{x^2-x-12}{2x}:\dfrac{x^2-2x-8}{4x^2}=\dfrac{x^2-x-12}{2x}\cdot\dfrac{4x^2}{x^2-2x-8}\\\\=\dfrac{(x+3)(x-4)}{2x}\cdot\dfrac{2x\cdot2x}{(x+2)(x-4)}=\dfrac{(x+3)}{1}\cdot\dfrac{2x}{(x+2)}\\\\=\boxed{\dfrac{2x^2+6x}{x+2}} [/tex]
