a) \(\displaystyle{{{x^2} - 5x + 6} \over {{x^2} + 7x + 12}}:{{{x^2} - 4x + 4} \over {{x^2} + 3x}}\)\(\displaystyle = {{{x^2} - 5x + 6} \over {{x^2} + 7x + 12}}.{{{x^2} + 3x} \over {{x^2} - 4x + 4}}\)
\(\displaystyle = {{{x^2} - 2x -3x+ 6} \over {{x^2} + 3x +4x + 12}}.{{x\left(x+3\right)} \over {\left( x-2\right)^2}}\)
\(\displaystyle = {{{x\left( {x - 2} \right) - 3\left( {x - 2} \right)}} \over {{x\left( {x + 3} \right) + 4\left( {x + 3} \right)}}}.{{x\left(x+3\right)} \over {\left( x-2\right)^2}}\)
\(=\dfrac{\left( {x - 2} \right)\left( {x - 3} \right)}{\left( {x + 3} \right)\left( {x + 4} \right)}.\) \(\displaystyle {{x\left(x+3\right)} \over {\left( x-2\right)^2}} \)
\(\displaystyle = {{x\left( {x - 2} \right)\left( {x - 3} \right)\left( {x + 3} \right)} \over {\left( {x + 3} \right)\left( {x + 4} \right){{\left( {x - 2} \right)}^2}}} \)\(\displaystyle= {{x\left( {x - 3} \right)} \over {\left( {x + 4} \right)\left( {x - 2} \right)}}\)
b) \(\displaystyle{{{x^2} + 2x - 3} \over {{x^2} + 3x - 10}}:{{{x^2} + 7x + 12} \over {{x^2} - 9x + 14}}\)\(\displaystyle = {{{x^2} + 2x - 3} \over {{x^2} + 3x - 10}}.{{{x^2} - 9x + 14} \over {{x^2} + 7x + 12}}\)
\(=\dfrac{{{x^2} + 3x - x - 3}}{{x^2} +5x - 2x - 10}.\) \(\dfrac{{x^2} - 7x - 2x + 14}{{x^2} + 3x + 4x + 12}\)
\(=\dfrac{x\left( {x + 3} \right) - \left( {x + 3} \right)}{{x\left( {x + 5} \right) - 2\left( {x + 5} \right)}}.\) \(\dfrac{{x\left( {x - 7} \right) - 2\left( {x - 7} \right)} }{{x\left( {x + 3} \right) + 4\left( {x + 3} \right)}}\)
\(=\dfrac{\left( {x + 3} \right)\left( {x - 1} \right)}{\left( {x + 5} \right)\left( {x - 2} \right)}.\) \(\dfrac{\left( {x - 7} \right)\left( {x - 2} \right)}{\left( {x + 3} \right)\left( {x + 4} \right)}\)
\(\displaystyle = {{\left( {x + 3} \right)\left( {x - 1} \right)\left( {x - 7} \right)\left( {x - 2} \right)} \over {\left( {x + 5} \right)\left( {x - 2} \right)\left( {x + 3} \right)\left( {x + 4} \right)}} \)\(\displaystyle= {{\left( {x - 1} \right)\left( {x - 7} \right)} \over {\left( {x + 5} \right)\left( {x + 4} \right)}} \)
