\(A=\left(\dfrac{1+2x}{4+2x}-\dfrac{x}{3x-6}+\dfrac{2x^2}{12-3x^2}\right).\dfrac{24-12x}{6+13x}\)\(=\left(\dfrac{1+2x}{2\left(x+2\right)}-\dfrac{x}{3\left(x-2\right)}-\dfrac{2x^2}{3\left(x-2\right)\left(x+2\right)}\right).\dfrac{-12\left(x-2\right)}{13x+6}\)\(=\left(\dfrac{3\left(1+2x\right)\left(x-2\right)}{6\left(x+2\right)\left(x-2\right)}-\dfrac{2x\left(x+2\right)}{6\left(x+2\right)\left(x-2\right)}-\dfrac{4x^2}{6x\left(x+2\right)\left(x-2\right)}\right).\dfrac{-2\left(x-2\right)}{13x+6}\)\(=\dfrac{6x^2-9x-6-2x^2-4x-4x^2}{6\left(x+2\right)\left(x-2\right)}.\dfrac{-12\left(x-2\right)}{13x+6}\)\(=\dfrac{-\left(13x+6\right)}{6\left(x+2\right)\left(x-2\right)}.\dfrac{-12\left(x-2\right)}{13x+6}\)
\(=\dfrac{2}{x+2}\)
