\(\dfrac{1}{x^2-4x+3}-\dfrac{1}{x-3}-\dfrac{1}{x-1}\)
\(=\dfrac{1}{x^2-3x-x+3}-\dfrac{1}{x-3}-\dfrac{1}{x-1}\)
\(=\dfrac{1}{x\left(x-3\right)-\left(x-3\right)}-\dfrac{1}{x-3}-\dfrac{1}{x-1}\)
\(=\dfrac{1}{\left(x-3\right)\left(x-1\right)}-\dfrac{1}{x-3}-\dfrac{1}{x-1}\)
\(=\dfrac{1}{\left(x-3\right)\left(x-1\right)}-\dfrac{1\left(x-1\right)}{\left(x-3\right)\left(x-1\right)}-\dfrac{1\left(x-3\right)}{\left(x-3\right)\left(x-1\right)}\)
\(=\dfrac{1-x+1-x+3}{\left(x-3\right)\left(x-1\right)}\)
\(=\dfrac{-x-x+1+1+3}{\left(x-1\right)\left(x+3\right)}\)
\(=\dfrac{-2x+5}{\left(x-1\right)\left(x-3\right)}\)
\(\dfrac{1}{x^3+8}-\dfrac{1}{x+2}-\dfrac{1}{x^2-2x+4}\)
\(=\dfrac{1}{\left(x+2\right)\left(x^2-2x+4\right)}-\dfrac{1}{x+2}-\dfrac{1}{x^2-2x+4}\)
\(=\dfrac{1}{\left(x+2\right)\left(x^2-2x+4\right)}-\dfrac{1\left(x^2-2x+4\right)}{\left(x+2\right)\left(x^2-2x+4\right)}-\dfrac{1\left(x+2\right)}{\left(x+2\right)\left(x^2-2x+4\right)}\)
\(=\dfrac{1-x^2+2x-4-x-2}{\left(x+2\right)\left(x^2-2x+4\right)}\)
\(=\dfrac{-x^2+2x-x+1-4-2}{\left(x+2\right)\left(x^2-2x+4\right)}\)
\(=\dfrac{-x^2-x-5}{\left(x+2\right)\left(x^2-2x+4\right)}\)
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