$a)$  

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

$x^2+3x+2\ne 0$  

$\Delta =9-8=1$  

$\sqrt{\Delta }=1$  

$x_1=\frac{-3-1}{2}=-2$  

$x_2=\frac{-3+1}{2}=-1$  

$x\notin \left\{-2;-1\right\}$  

 

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

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

 

$b)$  

$U\left(x\right)=\frac{x^3-3x^2-9x+27}{x^2-6x+9}$  

$x^2-6x+9\ne 0$  

${\left(x-3\right)}^2\ne 0$  

$x-3\ne 0\Rightarrow x\ne 3$  

$x\notin \left\{3\right\}$

 

$U\left(x\right)=\frac{x^3-3x^2-9x+27}{x^2-6x+9}=\frac{x^2\left(x-3\right)-9\left(x-3\right)}{{\left(x-3\right)}^2}=$  

$=\frac{\left(x^2-9\right)\left(x-3\right)}{{\left(x-3\right)}^2}=\frac{\left(x-3\right)\left(x+3\right)}{x-3}=x+3$  

 

$c)$  

$U\left(x\right)=\frac{x^3+7x^2-x-7}{x^2+6x-7}$  

$x^2+6x-7\ne 0$  

$\Delta =36+28=64$  

$\sqrt{\Delta }=8$  

$x_1=\frac{-6-8}{2}=-7$  

$x_2=\frac{-6+8}{2}=1$  

$x\notin \left\{-7;1\right\}$  

 

$U\left(x\right)=\frac{x^3+7x^2-x-7}{x^2+6x-7}=\frac{x^2\left(x+7\right)-\left(x+7\right)}{\left(x+7\right)\left(x-1\right)}=$   

$=\frac{\left(x^2-1\right)\left(x+7\right)}{\left(x+7\right)\left(x-1\right)}=\frac{\left(x-1\right)\left(x+1\right)}{x-1}=x+1$  

 

$d)$  

$U\left(x\right)=\frac{x^3+3x^2-25x-75}{x^2+8x+15}$  

$x^2+8x+15\ne 0$  

$\Delta =64-60=4$  

$\sqrt{\Delta }=2$  

$x_1=\frac{-8-2}{2}=-5$  

$x_2=\frac{-8+2}{2}=-3$  

$x\notin \left\{-5;-3\right\}$  

 

$U\left(x\right)=\frac{x^3+3x^2-25x-75}{x^2+8x+15}=\frac{x^2\left(x+3\right)-25\left(x+3\right)}{\left(x+5\right)\left(x+3\right)}=$  

$=\frac{\left(x^2-25\right)\left(x+3\right)}{\left(x+5\right)\left(x+3\right)}=\frac{\left(x-5\right)\left(x+5\right)}{x+5}=x-5$  

 

$e)$  

$U\left(x\right)=\frac{4x^3-4x^2-9x+9}{2x^2-5x+3}$  

$2x^2-5x+3\ne 0$  

$\Delta =25-24=1$  

$\sqrt{\Delta }=1$  

$x_1=\frac{5-1}{4}=1$  

$x_2=\frac{6}{4}=\frac{3}{2}$  

$x\notin \left\{1;\frac{3}{2}\right\}$    

 

$U\left(x\right)=\frac{4x^3-4x^2-9x+9}{2x^2-5x+3}=\frac{4x^2\left(x-1\right)-9\left(x-1\right)}{2\left(x-1\right)\left(x-\frac{3}{2}\right)}=$  

$=\frac{\left(4x^2-9\right)\left(x-1\right)}{2\left(x-1\right)\left(x-\frac{3}{2}\right)}=\frac{\left(2x-3\right)\left(2x+3\right)}{2\left(x-\frac{3}{2}\right)}=\frac{2\left(x-\frac{3}{2}\right)\left(2x+3\right)}{2\left(x-\frac{3}{2}\right)}=2x+3$     

 

$f)$  

$U\left(x\right)=\frac{25x^3+50x^2-x-2}{5x^2+9x-2}$  

$5x^2+9x-2\ne 0$  

$\Delta =81+40=121$  

$\sqrt{\Delta }=11$  

$x_1=\frac{-9-11}{10}=-2$  

$x_2=\frac{-9+11}{10}=\frac{1}{5}$  

$x\notin \left\{-2;\frac{1}{5}\right\}$  

 

$U\left(x\right)=\frac{25x^3+50x^2-x-2}{5x^2+9x-2}=\frac{25x^2\left(x+2\right)-1\left(x+2\right)}{5\left(x+2\right)\left(x-\frac{1}{5}\right)}=$  

$=\frac{\left(25x^2-1\right)\left(x+2\right)}{\left(x+2\right)\left(5x-1\right)}=\frac{\left(5x-1\right)\left(5x+1\right)}{5x-1}=5x+1$