6:
\(\dfrac{2}{x-3}+\dfrac{2x}{x^2-4x+3}+\dfrac{x}{x-1}\)
\(=\dfrac{2\left(x-1\right)+2x+x\left(x-3\right)}{\left(x-3\right)\left(x-1\right)}\)
\(=\dfrac{2x-2+x^2-3x+2x}{\left(x-3\right)\left(x-1\right)}=\dfrac{x^2+x-2}{\left(x-3\right)\left(x-1\right)}\)
\(=\dfrac{\left(x+2\right)\left(x-1\right)}{\left(x-3\right)\left(x-1\right)}=\dfrac{x+2}{x-3}\)
7: \(=\dfrac{\left(x+3\right)\left(x-3\right)+x^2-9}{x\left(x-3\right)}=\dfrac{x^2-9+x^2-9}{x\left(x-3\right)}\)
\(=\dfrac{2\left(x-3\right)\left(x+3\right)}{x\left(x-3\right)}=\dfrac{2x+6}{x}\)
8: \(=\dfrac{2\left(6x-5\right)}{\left(x-5\right)^2}+\dfrac{3x}{x-5}+\dfrac{x-2}{x+5}\)
\(=\dfrac{2\left(6x-5\right)\left(x-5\right)+3x\left(x^2-25\right)+\left(x-2\right)\left(x^2-10x+25\right)}{\left(x-5\right)^2\left(x+5\right)}\)
\(=\dfrac{2\left(6x^2-35x+25\right)+3x^3-7x+x^3-10x^2+25x-2x^2+20x-50}{\left(x-5\right)^2\cdot\left(x+5\right)}\)
\(=\dfrac{4x^3-12x^2+38x-50+12x^2-70x+50}{\left(x-5\right)^2\left(x+5\right)}\)
\(=\dfrac{4x^3-32x}{\left(x-5\right)^2\left(x+5\right)}\)
10: \(=\dfrac{6x+5x\left(x+3\right)+x\left(x-3\right)}{\left(x-3\right)\left(x+3\right)}\)
\(=\dfrac{6x+5x^2+15x+x^2-3x}{\left(x-3\right)\left(x+3\right)}=\dfrac{6x^2+18x}{\left(x-3\right)\left(x+3\right)}=\dfrac{6x}{x-3}\)
