\(\dfrac{x^2+x-6}{x^3-4x^2-18x+9}\)
\(=\dfrac{\left(x+3\right)\left(x-2\right)}{x^3+3x^2-7x^2-21x+3x+9}\)
\(=\dfrac{\left(x+3\right)\left(x-2\right)}{x^2\left(x+3\right)-7x\left(x+3\right)+3\left(x+3\right)}\)
\(=\dfrac{\left(x+3\right)\left(x-2\right)}{\left(x+3\right)\left(x^2-7x+3\right)}=\dfrac{x-2}{x^2-7x+3}\)
\(\dfrac{x^2+x-6}{x^3-4x^2-18x+9}\\ =\dfrac{\left(x^2+3x\right)-\left(2x+6\right)}{\left(x^3+3x^2\right)+\left(-7x^2-21x\right)+\left(3x+9\right)}\\ =\dfrac{x\left(x+3\right)-2\left(x+3\right)}{x^2\left(x+3\right)-7x\left(x+3\right)+3\left(x+3\right)}\\ =\dfrac{\left(x+3\right)\left(x-2\right)}{\left(x+3\right)\left(x^2-7x+3\right)}\\ =\dfrac{x-2}{x^2-7x+3}\)
