\(\dfrac{x}{x-2}+\dfrac{2-x}{x+2}+\dfrac{6-5x}{x^2-4}\\ =\dfrac{x\left(x+2\right)}{\left(x-2\right)\left(x+2\right)}+\dfrac{\left(2-x\right)\left(x-2\right)}{\left(x-2\right)\left(x+2\right)}+\dfrac{6-5x}{\left(x-2\right)\left(x+2\right)}\)
\(=\dfrac{x^2+2x}{\left(x-2\right)\left(x+2\right)}-\dfrac{\left(x-2\right)^2}{\left(x-2\right)\left(x+2\right)}+\dfrac{6-5x}{\left(x-2\right)\left(x+2\right)}\)
\(=\dfrac{x^2+2x+6-5x}{\left(x-2\right)\left(x+2\right)}-\dfrac{x^2-4x+4}{\left(x-2\right)\left(x+2\right)}\)
\(=\dfrac{x^2-3x+6-x^2+4x-4}{\left(x-2\right)\left(x+2\right)}\)
\(=\dfrac{x+2}{\left(x-2\right)\left(x+2\right)}\)
\(=\dfrac{1}{x-2}\)
\(=\dfrac{x^2+2x-x^2+4x-4+6-5x}{\left(x-2\right)\left(x+2\right)}=\dfrac{1}{x-2}\)
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