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