Simplify the following functions:

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

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 2)\ \    \frac{x^2-16}{{\left(x+4\right)}^2}=\frac{\left(x-4\right)\left(x+4\right)}{\left(x+4\right)\left(x+4\right)}=\frac{x-4}{x+4}    

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 3)\ \    \frac{a^2-b^2}{a^2b-{ab}^2}=\ \frac{\left(a-b\right)(a+b)}{ab(a-b)}=\frac{(a+b)}{ab}    

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

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

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  6)\ \   \frac{x^2-2x}{x^2+3x-10}=\frac{x\left(x-2\right)}{\left(x+5\right)\left(x-2\right)}=\frac{x}{\left(x+5\right)}    

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 7)\ \    \frac{x+3}{x^2-x-12}=\frac{x+3}{\left(x-4\right)\left(x+3\right)}=\frac{1}{x-4}    

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 8)\ \    \frac{x^2+2x-15}{x^2+4x-21}=\frac{\left(x+5\right)\left(x-3\right)}{\left(x+7\right)\left(x-3\right)}=\frac{x+5}{x+7}    

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

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 10)\ \    \frac{x^3-{xy}^2}{x^3+{2x}^2y+{xy}^2}=\frac{x\left(x-y\right)\left(x+y\right)}{x^2\left(x+y\right)+2y\left(x+y\right)}=\frac{x\left(x-y\right)\left(x+y\right)}{x\left(x+y\right)\left(x+y\right)}=\frac{x-y}{x+y}    

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  11)\ \   \frac{{2x}^2+7x+3}{{3x}^2+10x+3}=\frac{{2x}^2+x+6x+3}{3x^2+9x+x+3}    

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

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 12)\ \    \frac{{2x}^2+x-3}{{3x}^2+x-4}=\frac{2x^2-2x+3x-3}{3x^2+4x-3x-4}    

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

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  13)\ \   \frac{x^2-16}{x^2-9}\times{}\frac{x+3}{x+4}=\frac{\left(x-4\right)\left(x+4\right)\left(x+3\right)}{\left(x-3\right)\left(x+3\right)\left(x+4\right)}=\frac{x-4}{x-3}    

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 14)\ \    \frac{a^2-{ab}^2}{a^2-{16b}^2}\times{}\frac{a+4b}{a-3b}=\frac{\left(a-3b\right)(a+3b(a+4b)}{\left(a-4b\right)\left(a+4b\right)\left(a-3b\right)}=\frac{a+3b}{a-4b}    

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 15)\ \    \frac{x^2+6x+5}{x^2+5x}\times{}\frac{x^3-x}{x^2-1}=\frac{\left(x+1\right)\left(x+5\right)x\left(x-1\right)\left(x+1\right)}{x\left(x+5\right)\left(x-1\right)\left(x+1\right)}=(x+1)    

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  16)\ \   \frac{x+y}{x^2-xy}\div{}\frac{x^2+xy}{x-y}=\frac{x+y}{x(x-y)}\times{}\frac{x-y}{x(x+y)}=\frac{1}{x^2}    

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 17)\ \    \frac{x^2-5x}{2x-3y}\div{}\frac{x^2-25}{{4x}^2-9y^2}=\frac{x\left(x-5\right)}{\left(2x-3y\right)}\times{}\frac{\left(2x-3y\right)\left(2x+3y\right)}{\left(x-5\right)\left(x+5\right)}=\frac{x(2x+3y)}{(x+5)}    

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  18)\ \   \frac{x^2+2-6}{x^2+2x-3}\div{}\frac{x^2+5x-14}{x^2+4x-5}=\frac{\left(x+3\right)\left(x-2\right)}{\left(x+3\right)\left(x-1\right)}\times{}\frac{\left(x+5\right)\left(x-1\right)}{\left(x+7\right)\left(x-2\right)}=\frac{x+5}{x+7}    

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  19)\ \   \frac{{3x}^2-x-2}{x^2+x-2}\div{}\frac{{3x}^2-7x-6}{x^2-x-6}    

     =\frac{{3x}^2-3x+2x-2}{x^2+2x-x-2}\times{}\frac{x^2-3x+2x-6}{3x^2-9x+2x-6}    

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

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

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 20)\ \   \frac{{2x}^2-3x-2}{x^2+7x+12}\times{}\frac{x^2+x-12}{x^2+3x-10}\div{}\frac{{2x}^2-5x-3}{x^2+8x+15}    

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

     =\frac{2x\left(x-2\right)+\left(x-2\right)}{\left(x+4\right)\left(x+3\right)}\times{}\frac{\left(x+4\right)\left(x-3\right)}{\left(x+5\right)\left(x-2\right)}\times{}\frac{\left(x+3\right)\left(x+5\right)}{\left(2x+1\right)\left(x-3\right)}=1   

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  21)\ \   \frac{2x+3}{3}-\frac{2x-4}{4}   

     =\frac{4\left(2x+3\right)-3\left(2x-4\right)}{12}   

     =\frac{8x+12-6x+12}{12}=\frac{2x+24}{12}=\frac{x+12}{6}   

     =\frac{3}{x-1}-\frac{3}{x+1}   

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

     =\frac{3x+3-3x+3}{\left(x-1\right)(x+1)}=\frac{6}{\left(x-1\right)(x+1)}   

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   22)\ \  \frac{3x-1}{4x}+\frac{3-5x}{12x}   

     =\frac{12x\left(3x-1\right)+4x\left(3-5x\right)}{48x^2}   

     =\frac{3}{x-1}-\frac{3}{x+1}   

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

     =\frac{3x+3-3x+3}{\left(x-1\right)(x+1)}=\frac{6}{\left(x-1\right)(x+1)}   

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  23)\ \  \frac{3x-1}{4x}+\frac{3-5x}{12x}   

    =\frac{12x\left(3x-1\right)+4x\left(3-5x\right)}{48x^2}   

    =\frac{36x^2-12x+12x-20x^2}{48x^2}=\frac{{16x}^2}{48x^2}=\frac{1}{3}   

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

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

    =\frac{x}{\left(x-1\right)(x+1)}   

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  25)\ \  \frac{6}{a+b}-\frac{5}{a-b}+\frac{11b}{a^2-b^2}   

    =\frac{6\left(a-b\right)-5\left(a+b\right)+11b}{\left(a-b\right)\left(a+b\right)}   

    =\frac{6a-6b-5a-5b+11b}{\left(a-b\right)(a+b)}   

    =\frac{a}{\left(a-b\right)\left(a+b\right)}   

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 26)\ \   \frac{6}{x-2}+\frac{8}{2x-4}   

    =\frac{6\left(2x-4\right)+8\left(x-2\right)}{\left(x-2\right)\left(2x-4\right)}=\frac{12x-24+8x-16}{\left(x-2\right)\left(2x-4\right)}   

   =\frac{20x-40}{\left(x-2\right)\left(4-4\right)}=\frac{20\left(x-2\right)}{\left(x-2\right)\left(x-4\right)}=\frac{20}{x-4}  

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 27)\ \  \frac{1}{x^2-11x+30}+\frac{1}{x^2-9x+20}  

   =\frac{1}{\left(x-6\right)\left(x-5\right)}+\frac{1}{\left(x-5\right)\left(x-4\right)}  

   =\frac{\left(x-4\right)+\left(x-6\right)}{\left(x-6\right)\left(x-5\right)\left(x-4\right)}  

   =\frac{2(x-5)}{\left(x-6\right)\left(x-5\right)\left(x-4\right)}=\frac{2}{\left(x-6\right)(x-4)}  

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 28)\ \  \frac{1}{x^2-8x+15}+\frac{1}{x^2-4x+3}-\frac{2}{\left(x^2-6x+5\right)}  

   =\frac{1}{\left(x-3\right)\left(x-5\right)}+\frac{1}{\left(x-3\right)\left(x-1\right)}-\frac{2}{\left(x-3\right)\left(x-2\right)}  

   =\frac{\left(x-1\right)\left(x-2\right)+\left(x-2\right)\left(x-5\right)-2\left(x-1\right)\left(x-5\right)}{\left(x-1\right)\left(x-2\right)\left(x-3\right)\left(x-5\right)}  

   =\frac{x^2-3x+2+x^2-7x+10-2x^2+12x-10}{\left(x-1\right)\left(x-2\right)\left(x-3\right)\left(x-5\right)}  

   =\frac{2(x+1}{\left(x-1\right)\left(x-2\right)\left(x-3\right)(x-5)}  

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 29)\ \  \frac{x-3}{x^2-7x+12}+\frac{2\left(x-1\right)}{x^2-4x+3}-\frac{3\left(x-4\right)}{x^2-5x+4}  

m38x   =\frac{x-3}{\left(x-3\right)\left(x-4\right)}+\frac{2\left(x-1\right)}{\left(x-3\right)\left(x-1\right)}-\frac{3\left(x-4\right)}{\left(x-4\right)\left(x-1\right)}  

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

   =\frac{x^2-4x+3+2x^2-10x+8-{3x}^2+21x-36}{\left(x-1\right)\left(x-3\right)\left(x-4\right)}  

   =\frac{7x-25}{\left(x-1\right)\left(x-3\right)(x-4)}  

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 30)\ \ \frac{x^2-x-6}{x^2-9}+\frac{x^2+2x-24}{x^2-x-12}  

  =\frac{\left(x-6\right)(x+1)}{\left(x-3\right)(x+3)}+\frac{\left(x+6\right)(x-4)}{\left(x-4\right)(x+3)}  

  =\frac{\left(x-6\right)\left(x+1\right)}{\left(x-3\right)\left(x+3\right)}+\frac{\left(x+6\right)}{\left(x+3\right)}  

  =\frac{(x^2-5x-6)+(x^2+3x-18)}{\left(x-3\right)\left(x+3\right)}  

  =\frac{{2x}^2-2x-24}{\left(x-3\right)(x+3)}=\frac{2\left(x-4\right)(x+3)}{\left(x-3\right)(x+3)}=\frac{2(x-4)}{(x-3)}  

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 31)\ \ \left(\frac{2x}{4x+1}-\frac{x}{2x+3}\right)\div{}\ \frac{10x+5}{2x+3}  

  =\frac{4x^2+6x-{4x}^2-x}{\left(4x+1\right)\left(2x+3\right)}\times{}\frac{2x+3}{5\left(2x+1\right)}  

  =\frac{5x(2x+3)}{5\left(4x+1\right)\left(2x+3\right)(2x+1)}=\frac{x}{\left(4x+1\right)(2x+1)}  

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  32)\ \ \left(\frac{a^2+a-12}{a^2+6a+8}+\frac{a^2-6a+5}{a^2-3a-10}\right)\ of\ \frac{a^2+8+12}{a^2+4a-12}\  
  =\left(\frac{\left(a+4\right)\left(a+3\right)}{\left(a+4\right)\left(a+2\right)}+\frac{\left(a-5\right)\left(a-1\right)}{\left(a-5\right)\left(a+2\right)}\right)\times{}\frac{(a+2)(a+6)}{(a+6)(a-2)}  

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

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