a. \(\displaystyle {\displaystyle {{x \over {x - 1}} - {{x + 1} \over x}} \over {\displaystyle {x \over {x + 1}} - {{x - 1} \over x}}}\)
\(\displaystyle = \left( {{x \over {x - 1}} - {{x + 1} \over x}} \right)\)\(:\displaystyle \left( {{x \over {x + 1}} - {{x - 1} \over x}} \right)\)
\(\displaystyle = {{{x^2} - \left( {x + 1} \right)\left( {x - 1} \right)} \over {x\left( {x - 1} \right)}}\)\(:\displaystyle {{{x^2} - \left( {x - 1} \right)\left( {x + 1} \right)} \over {x\left( {x + 1} \right)}} \)
\(\displaystyle = {1 \over {x\left( {x - 1} \right)}}.{{x\left( {x + 1} \right)} \over 1} = {{x + 1} \over {x - 1}}\)
b. \(\displaystyle {\displaystyle {{5 \over 4} - {5 \over {x + 1}}} \over {\displaystyle {{9 - {x^2}} \over {{x^2} + 2x + 1}}}}\)
\( \displaystyle = \left( {{5 \over 4} - {5 \over {x + 1}}} \right):\left( {{{9 - {x^2}} \over {{x^2} + 2x + 1}}} \right)\)
\(\displaystyle = {{5\left( {x + 1} \right) - 20} \over {4\left( {x + 1} \right)}}:{{\left( {3 + x} \right)\left( {3 - x} \right)} \over {{{\left( {x + 1} \right)}^2}}}\)
\( \displaystyle = {{5\left( {x - 3} \right)} \over {4\left( {x + 1} \right)}}.{{{{\left( {x + 1} \right)}^2}} \over {\left( {3 + x} \right)\left( {3 - x} \right)}}\)
\(\displaystyle = {{ - 5\left( {3 - x} \right)\left( {x + 1} \right)} \over {4\left( {3 + x} \right)\left( {3 - x} \right)}} = {{ - 5\left( {x + 1} \right)} \over {4\left( {3 + x} \right)}}\)
