\(\dfrac{2x^2+4x}{x^3-4x}+\dfrac{x^2-4}{x^2+2x}+\dfrac{2}{2-x}\)
\(=\dfrac{2x^2+4x}{x\left(x-2\right)\left(x+2\right)}+\dfrac{x^2-4}{x\left(x+2\right)}-\dfrac{2}{x-2}\)\(=\dfrac{2x^2+4x+\left(x^2-4\right)\left(x-2\right)-2x\left(x+2\right)}{x\left(x-2\right)\left(x+2\right)}\)
\(=\dfrac{2x^2+4x+x^3-2x^2-4x+8-2x^2-4x}{x\left(x-2\right)\left(x+2\right)}\)
\(=\dfrac{x^3-2x^2-4x+8}{x\left(x-2\right)\left(x+2\right)}\)
\(=\dfrac{\left(x^3+8\right)-\left(2x^2+4x\right)}{x\left(x-2\right)\left(x+2\right)}\)
\(=\dfrac{\left(x+2\right)\left(x^2-2x+4\right)-2x\left(x+2\right)}{x\left(x-2\right)\left(x+2\right)}\)
\(=\dfrac{\left(x+2\right)\left(x^2-2x+4-2x\right)}{x\left(x-2\right)\left(x+2\right)}\)
\(=\dfrac{\left(x+2\right)\left(x^2-4x+4\right)}{x\left(x-2\right)\left(x+2\right)}\)
\(=\dfrac{\left(x+2\right)\left(x-2\right)^2}{x\left(x-2\right)\left(x+2\right)}\)
\(=\dfrac{x-2}{x}.\)
