$\displaystyle \begin{array}{{>{\displaystyle}l}} \frac{x^{3} +x^{2} -6x}{x^{3} -4x} \ \\ =\frac{x\left( x^{2} +x-6\right)}{x\left( x^{2} -4\right)} =\frac{x\left( x^{2} +3x-2x-6\right)}{x( x+2)( x-2)}\\ =\frac{x[ x( x+3) -2( x+3)]}{x( x+2)( x-2)} =\frac{x( x+3)( x-2)}{x( x+2)( x-2)}\\ =\frac{x+3}{x+2} \ \\ b)\frac{x^{2} -3x+2}{x^{3} -1} =\frac{x^{2} -x-2x+2}{( x-1)\left( x^{2} +x+1\right)}\\ =\frac{x( x-1) -2( x-1)}{( x-1)\left( x^{2} +x+1\right)} =\frac{( x-1)( x-2)}{( x-1)\left( x^{2} +x+1\right)}\\ =\frac{x-2}{x^{2} +x+1}\\ e)\frac{2x^{2} +6x}{x^{3} +7x^{2} +12x} =\frac{2x( x+3)}{x\left( x^{2} +7x+12\right)}\\ =\frac{2x( x+3)}{x[ x( x+4) +3( x+4)]} =\frac{2x( x+3)}{x( x+4)( x+3)}\\ =\frac{2}{x+4} \ \\ c)\frac{x^{2} -25}{5x-x^{2}} =\frac{( x-5)( x+5)}{x( 5-x)} =\frac{-x-5}{x} \ \\ f)\frac{x^{2} +4x-4}{2x^{2} -4x} =\frac{( x-2)^{2}}{2x( x-2)} =\frac{x-2}{2x} \ \\ g)\frac{x^{3} -1}{x^{3} -3x^{2} +3x-1} =\frac{( x-1)\left( x^{2} +x+1\right)}{( x-1)^{2}}\\ =\frac{x^{2} +x+1}{x-1} \ \\ d)\frac{y^{2} -xy}{4xy-4y^{2}} =\frac{y( y-x)}{4y( x-y)} =\frac{-1}{4} \ \\ h)\frac{2x^{2} -3x-2}{2x^{2} +5x+2} =\frac{2x^{2} +x-4x-2}{2x^{2} +4x+x+2}\\ = \ \frac{x( 2x+1) -2( 2x+1)}{2x( x+2) +( x+2)} =\frac{( 2x+1)( x-2)}{( x+2)( 2x+1)}\\ =\frac{x-2}{x+2}\\ \end{array}$
