\[\begin{array}{l}
{x^2} - 3{x^2} - 4x + 12 = - 2{x^2} - 4x + 12\\
= - 2\left( {{x^2} + 2x - 6} \right) = - 2\left[ {\left( {{x^2} + 2x + 1} \right) - 7} \right]\\
= - 2\left[ {{{\left( {x + 1} \right)}^2} - 7} \right] = - 2\left( {x + 1 - \sqrt 7 } \right)\left( {x + 1 + \sqrt 7 } \right).\\
9{x^2} - 36 = \left( {3x - 6} \right)\left( {3x + 6} \right).\\
{x^3} - x + 3{x^2}y + 3x{y^2} + {y^3} - y\\
= \left( {{x^3} + 3{x^2}y + 3x{y^2} + {y^3}} \right) - \left( {x + y} \right)\\
= {\left( {x + y} \right)^3} - \left( {x + y} \right)\\
= \left( {x + y} \right)\left[ {{{\left( {x + y} \right)}^2} - 1} \right]\\
= \left( {x + y} \right)\left( {x + y - 1} \right)\left( {x + y + 1} \right)\\
a{x^2} + ay - b{x^2} - by\\
= {x^2}\left( {a - b} \right) + y\left( {a - b} \right)\\
= \left( {a - b} \right)\left( {{x^2} + y} \right).
\end{array}\]
