a) [tex] \frac{(4 x^{2} +13x+9)}{(4 x^{2} +5x-9)} = \frac{4 x^{2} +9x+4x+9}{4 x^{2} +9x-4x-9}= \frac{x*(4x+9)+4x+9}{x*(4x+9)-(4x+9)}= \frac{(x+1)*(4x+9)}{(x-1)*(4x+9)}= \frac{x+1}{x-1} [/tex]
b) [tex] \frac{ ( x^{3} +3 x^{2} -x-3)}{( x^{3} - x^{2} -9x+9)} = \frac{ x^{2} *(x+3)-(x+3)}{ x^{2} *(x-1)-9(x-1)}= \frac{ (x^{2}-1)*(x+3) }{( x^{2} -9)*(x-1)}= \frac{(x-1)*(x+1)*(x+3)}{(x-3)*(x+3)*(x-1)} [/tex][tex]=\frac{x+1}{x-3} [/tex]
c) [tex] \frac{(2 x^{3} +5 x^{2} -8x-20)}{(4 x^{3} +8 x^{2} -25x-50)}= \frac{ x^{2} *(2x+5)-4(2x+5)}{4 x^{2} *(x+2)-25(x+2)}= \frac{( x^{2} -4)*(2x+5)}{(4 x^{2} -25)*(x+2)}[/tex][tex]=\frac{(x-2)*(x+2)*(2x+5)}{(2x-5)*(2x+5)*(x+2)}= \frac{x-2}{2x-5} [/tex]
d) [tex] \frac{( x^{3} +7 x^{2} -9x-63)}{ (x^{3} -2 x^{2} -9x+18)} = \frac{ x^{2} *(x+7)-9(x+7)}{ x^{2} *(x-2)-9(x-2)}= \frac{( x^{2} -9)*(x+7)}{( x^{2} -9)*(x-2)}= \frac{(x-3)*(x+3)*(x+7)}{(x-3)*(x+3)*(x-2)} [/tex][tex]= \frac{x+7}{x-2} [/tex]
