\(\dfrac{x-3}{x^2-2x}+\dfrac{2x-5}{x^2+4x+4}=\dfrac{x-3}{x\left(x-2\right)}+\dfrac{2x-5}{\left(x+2\right)^2}=\dfrac{\left(3-x\right)\left(x+2\right)}{x\left(x+2\right)\left(x+2\right)}+\dfrac{x\left(2x-5\right)}{x\left(x+2\right)^2}=\dfrac{3x+6-x^2-2x+2x^2-5x}{x\left(x+2\right)^2}=\dfrac{x^2-4x+6}{x\left(x+2\right)^2}\)
\(=\dfrac{x-3}{x\left(x-2\right)}+\dfrac{2x-5}{\left(x+2\right)\left(x+2\right)}\)
\(=\dfrac{\left(x-3\right)\left(x+2\right)\left(x+2\right)}{x\left(x-2\right)\left(x+2\right)\left(x+2\right)}+\dfrac{x\left(x-2\right)\left(2x-5\right)}{x\left(x-2\right)\left(x+2\right)\left(x+2\right)}\)
\(=\dfrac{x^3+2x^2-3x^2-6x+2x^2+4x-6x-12}{x\left(x-2\right)\left(x+2\right)\left(x+2\right)}+\dfrac{2x^3-5x^2-4x^2+10x}{x\left(x-2\right)\left(x+2\right)\left(x+2\right)}\)
\(=\dfrac{x^3+2x^2-3x^2-6x+2x^2+4x-6x-12+2x^3-5x^2-4x^2+10x}{x\left(x-2\right)\left(x+2\right)\left(x+2\right)}\)
\(=\dfrac{3x^3-8x^2+2x-12}{x\left(x-2\right)\left(x+2\right)\left(x+2\right)}\)
