Answer
$$x = - 2,{\text{ }}x = - 1,{\text{ }}x = 3,{\text{ }}x = 2$$
Work Step by Step
$$\eqalign{
& \left( {{x^2} + 3x + 2} \right)\left( {{x^2} - 5x + 6} \right) = 0 \cr
& {\text{Factoring each trinomial}} \cr
& {x^2} + 3x + 2 = \left( {x + 2} \right)\left( {x + 1} \right) \cr
& {x^2} - 5x + 6 = \left( {x - 3} \right)\left( {x - 2} \right) \cr
& {\text{Therefore}} \cr
& \left( {x + 2} \right)\left( {x + 1} \right)\left( {x - 3} \right)\left( {x - 2} \right) = 0 \cr
& {\text{Zero - factor property}} \cr
& x + 2 = 0,{\text{ }}x + 1 = 0,{\text{ }}x - 3 = 0,{\text{ }}x - 2 = 0 \cr
& {\text{Solving all the linear equations}} \cr
& x = - 2,{\text{ }}x = - 1,{\text{ }}x = 3,{\text{ }}x = 2 \cr} $$