a) \(\displaystyle P:{{4{x^2} - 16} \over {2x + 1}} = {{4{x^2} + 4x + 1} \over {x - 2}}\)
\(\displaystyle \Rightarrow P = {{4{x^2} + 4x + 1} \over {x - 2}}. {{4{x^2} - 16} \over {2x + 1}} \)
\(\displaystyle P = {{{{\left( {2x + 1} \right)}^2}} \over {x - 2}} .{{4\left( {x + 2} \right)\left( {x - 2} \right)} \over {2x + 1}}\)
\(\displaystyle P = 4\left( {x + 2} \right)\left( {2x + 1} \right) \)
\(\displaystyle P = 4\left( {2{x^2} + x + 4x + 2} \right) \)
\(\displaystyle P = 8{x^2} + 40x + 8 \)
b) \(\displaystyle{{2{x^2} + 4x + 8} \over {{x^3} - 3{x^2} - x + 3}}:P \) \(\displaystyle = {{{x^3} - 8} \over {\left( {x + 1} \right)\left( {x - 3} \right)}}\)
\(\displaystyle \Rightarrow P = {{2{x^2} + 4x + 8} \over {{x^3} - 3{x^2} - x + 3}}:\) \(\displaystyle {{{x^3} - 8} \over {\left( {x + 1} \right)\left( {x - 3} \right)}} \)
\(\displaystyle P = {{2\left( {{x^2} + 2x + 4} \right)} \over {\left( {x - 3} \right)\left( {x + 1} \right)\left( {x - 1} \right)}}.\) \(\displaystyle{{\left( {x + 1} \right)\left( {x - 3} \right)} \over {\left( {x - 2} \right)\left( {{x^2} + 2x + 4} \right)}} \)
\(\displaystyle P = {2 \over {\left( {x + 1} \right)\left( {x - 2} \right)}} = {2 \over {{x^2} - x - 2}} \)
