a) \(\displaystyle{{x - y} \over {{x^3} + {y^3}}}.Q = {{{x^2} - 2xy + {y^2}} \over {{x^2} - xy + {y^2}}}\)
\(\displaystyle \Rightarrow Q = {{{x^2} - 2xy + {y^2}} \over {{x^2} - xy + {y^2}}}:{{x - y} \over {{x^3} + {y^3}}} \)
\(\displaystyle Q= {{{{\left( {x - y} \right)}^2}} \over {{x^2} - xy + {y^2}}}.{{{x^3} + {y^3}} \over {x - y}} \)
\(\displaystyle Q = {{{{\left( {x - y} \right)}^2}\left( {x + y} \right)\left( {{x^2} - xy + {y^2}} \right)} \over {\left( {{x^2} - xy + {y^2}} \right)\left( {x - y} \right)}} \)
\(\displaystyle Q = \left( {x - y} \right)\left( {x + y} \right) = {x^2} - {y^2} \)
b) \(\displaystyle{{x + y} \over {{x^3} - {y^3}}}.Q = {{3{x^2} + 3xy} \over {{x^2} + xy + {y^2}}}\)
\(\displaystyle \Rightarrow Q = {{3{x^2} + 3xy} \over {{x^2} + xy + {y^2}}}:{{x + y} \over {{x^3} - {y^3}}} \)
\(\displaystyle Q= {{3{x^2} + 3xy} \over {{x^2} + xy + {y^2}}}.{{{x^3} - {y^3}} \over {x + y}} \)
\(\displaystyle Q = {{3x\left( {x + y} \right)\left( {x - y} \right)\left( {{x^2} + xy + {y^2}} \right)} \over {\left( {{x^2} + xy + {y^2}} \right)\left( {x + y} \right)}}\)
\(\displaystyle Q = 3x\left( {x - y} \right) = 3{x^2} - 3xy \)
