Reduce each of the following to the lowest terms:

Question 1:  $\frac{x^2+x}{x^2-1}$

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

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Question 2:  $\frac{x^2-16}{{ ( x+4 ) }^2}$

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

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Question 3:  $\frac{a^2-b^2}{a^2b-{ab}^2}$

$\frac{a^2-b^2}{a^2b-{ab}^2}$ $=$ $\frac{ ( a-b ) (a+b)}{ab(a-b)}$ $=$ $\frac{(a+b)}{ab}$

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Question 4:  $\frac{a^2+ab}{{2a}^3b-{2ab}^3}$

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

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Question 5:  $\frac{x^2-9}{x^2+x-6}$

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

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Question 6:  $\frac{x^2-2x}{x^2+3x-10}$

$\frac{x^2-2x}{x^2+3x-10}$ $=$ $\frac{x ( x-2 ) }{ ( x+5 ) ( x-2 ) }$ $=$ $\frac{x}{ ( x+5 ) }$

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Question 7:  $\frac{x+3}{x^2-x-12}$

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

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Question 8:  $\frac{x^2+2x-15}{x^2+4x-21}$

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

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Question 9:  $\frac{x^2-5x+4}{x^2-3x-4}$

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

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Question 10:  $\frac{x^3-{xy}^2}{x^3+{2x}^2y+{xy}^2}$

$\frac{x^3-{xy}^2}{x^3+{2x}^2y+{xy}^2}$ $=$ $\frac{x ( x-y ) ( x+y ) }{x^2 ( x+y ) +2y ( x+y ) }$ $=$ $\frac{x ( x-y ) ( x+y ) }{x ( x+y ) ( x+y ) }$ $=$ $\frac{x-y}{x+y}$

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Question 11:  $\frac{{2x}^2+7x+3}{{3x}^2+10x+3}$

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

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

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Question 12:  $\frac{{2x}^2+x-3}{{3x}^2+x-4}$

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

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

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Simplify:

Question 13:  $\frac{x^2-16}{x^2-9}$ $\times$ $\frac{x+3}{x+4}$

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

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Question 14:  $\frac{a^2-{ab}^2}{a^2-{16b}^2}$ $\times$ $\frac{a+4b}{a-3b}$

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

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Question 15:  $\frac{x^2+6x+5}{x^2+5x}$ $\times$ $\frac{x^3-x}{x^2-1}$

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

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Question 16:  $\frac{x+y}{x^2-xy}$ $\div$ $\frac{x^2+xy}{x-y}$

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

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Question 17:  $\frac{x^2-5x}{2x-3y}$ $\div$ $\frac{x^2-25}{{4x}^2-9y^2}$

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

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Question 18:  $\frac{x^2+2-6}{x^2+2x-3}$ $\div$ $\frac{x^2+5x-14}{x^2+4x-5}$

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

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Question 19:  $\frac{{3x}^2-x-2}{x^2+x-2}$ $\div$ $\frac{{3x}^2-7x-6}{x^2-x-6}$

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

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

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

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

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

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

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

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

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

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Simplify:

Question 21:  $\frac{2x+3}{3}-\frac{2x-4}{4}$

$\frac{2x+3}{3}-\frac{2x-4}{4}$

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

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

$=$ $\frac{3}{x-1}-\frac{3}{x+1}$

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

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

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Question 22:  $\frac{3x-1}{4x}+\frac{3-5x}{12x}$

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

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

$=$ $\frac{3}{x-1}-\frac{3}{x+1}$

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

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

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Question 23:  $\frac{3x-1}{4x}+\frac{3-5x}{12x}$

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

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

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

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Question 24:  $\frac{x}{x-1}-\frac{x^2}{x^2-1}$

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

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

$=$ $\frac{x}{ ( x-1 ) (x+1)}$

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Question 25:  $\frac{6}{a+b}-\frac{5}{a-b}+\frac{11b}{a^2-b^2}$

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

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

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

$=$ $\frac{a}{ ( a-b ) ( a+b ) }$

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Question 26:  $\frac{6}{x-2}+\frac{8}{2x-4}$

$\frac{6}{x-2}+\frac{8}{2x-4}$

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

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

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Question 27:  $\frac{1}{x^2-11x+30}+\frac{1}{x^2-9x+20}$

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

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

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

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

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Question 28:  $\frac{1}{x^2-8x+15}+\frac{1}{x^2-4x+3}-\frac{2}{ ( x^2-6x+5 ) }$

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

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

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

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

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

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Question 29:  $\frac{x-3}{x^2-7x+12}+\frac{2 ( x-1 ) }{x^2-4x+3}-\frac{3 ( x-4 ) }{x^2-5x+4}$

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

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

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

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

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

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Question 30:  $\frac{x^2-x-6}{x^2-9}+\frac{x^2+2x-24}{x^2-x-12}$

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

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

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

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

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

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Question 31:  $( \frac{2x}{4x+1}-\frac{x}{2x+3} ) \div{}\ \frac{10x+5}{2x+3}$

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

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

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

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Question 32:  $( \frac{a^2+a-12}{a^2+6a+8}+\frac{a^2-6a+5}{a^2-3a-10} )$ of $\frac{a^2+8+12}{a^2+4a-12}$

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