\(\begin{array}{l}
a)\,\dfrac{1}{{2{x^2} + 7x - 15}} = \dfrac{1}{{2{x^2} + 10x - 3x - 15}} = \dfrac{1}{{2x\left( {x + 5} \right) - 3\left( {x + 5} \right)}} = \dfrac{1}{{\left( {2x - 3} \right)\left( {x + 5} \right)}}\\
\dfrac{{x + 2}}{{{x^2} + 3x - 10}} = \dfrac{{x + 2}}{{{x^2} + 5x - 2x - 10}} = \dfrac{{x + 2}}{{x\left( {x + 5} \right) - 2\left( {x + 5} \right)}} = \dfrac{{x + 2}}{{\left( {x - 2} \right)\left( {x + 5} \right)}}\\
MTC:\,\left( {x + 5} \right)\left( {x - 2} \right)\left( {2x - 3} \right)\\
\Rightarrow \dfrac{1}{{2{x^2} + 7x - 15}} = \dfrac{{x - 2}}{{\left( {x + 5} \right)\left( {x - 2} \right)\left( {2x - 3} \right)}}\\
\dfrac{{x + 2}}{{{x^2} + 3x - 10}} = \dfrac{{\left( {x + 2} \right)\left( {2x - 3} \right)}}{{\left( {x + 5} \right)\left( {x - 2} \right)\left( {2x - 3} \right)}}\\
\dfrac{1}{{x + 5}} = \dfrac{{\left( {x - 2} \right)\left( {2x - 3} \right)}}{{\left( {x + 5} \right)\left( {x - 2} \right)\left( {2x - 3} \right)}}\\
b)\,\dfrac{1}{{ - {x^2} + 3x - 2}} = \dfrac{{ - 1}}{{{x^2} - 2x - x + 2}} = \dfrac{{ - 1}}{{x\left( {x - 2} \right) - \left( {x - 2} \right)}} = \dfrac{{ - 1}}{{\left( {x - 2} \right)\left( {x - 1} \right)}}\\
\dfrac{1}{{{x^2} + 5x - 6}} = \dfrac{1}{{{x^2} + 6x - x - 6}} = \dfrac{1}{{x\left( {x + 6} \right) - \left( {x + 6} \right)}} = \dfrac{1}{{\left( {x - 1} \right)\left( {x + 6} \right)}}\\
\dfrac{1}{{ - {x^2} + 4x - 3}} = \dfrac{{ - 1}}{{{x^2} - 3x - x + 3}} = \dfrac{{ - 1}}{{\left( {x - 1} \right)\left( {x - 3} \right)}}\\
MTC:\,\left( {x - 1} \right)\left( {x - 2} \right)\left( {x - 3} \right)\left( {x + 6} \right)\\
\Rightarrow \dfrac{1}{{ - {x^2} + 3x - 2}} = \dfrac{{ - \left( {x - 3} \right)\left( {x + 6} \right)}}{{\left( {x - 1} \right)\left( {x - 2} \right)\left( {x - 3} \right)\left( {x + 6} \right)}}\\
\dfrac{1}{{{x^2} + 5x - 6}} = \dfrac{{\left( {x - 2} \right)\left( {x - 3} \right)}}{{\left( {x - 1} \right)\left( {x - 2} \right)\left( {x - 3} \right)\left( {x + 6} \right)}}\\
\dfrac{1}{{ - {x^2} + 4x - 3}} = \dfrac{{ - \left( {x - 2} \right)\left( {x + 6} \right)}}{{\left( {x - 1} \right)\left( {x - 2} \right)\left( {x - 3} \right)\left( {x + 6} \right)}}
\end{array}\)
