\(\frac{3x^3-7x^2+5x+1}{2x^3-x^2-4x+3}\)
= \(\frac{\left(3x^2-4x+1\right)\left(x-1\right)}{\left(2x^2+x-3\right)\left(x-1\right)}\)
= \(\frac{\left(3x^2-x-3x+1\right)\left(x-1\right)}{\left(2x^2+3x-2x-3\right)\left(x-1\right)}\)
= \(\frac{\left[x\left(3x-1\right)-\left(3x-1\right)\right]\left(x-1\right)}{\left[x\left(2x+3\right)-\left(2x+3\right)\right]\left(x-1\right)}\)
= \(\frac{\left(3x-1\right)\left(x-1\right)\left(x-1\right)}{\left(2x+3\right)\left(x-1\right)\left(x-1\right)}\)
= \(\frac{\left(3x-1\right)\left(x-1\right)^2}{\left(2x+3\right)\left(x-1\right)^2}\)
= \(\frac{3x-1}{2x+3}\)