\(\left| {{x^2} + 4x + 2} \right| = x + 2.\)
Уравнение вида \(\left| {f\left( x \right)} \right| = g\left( x \right)\) равносильно системе: \(\left\{ {\begin{array}{*{20}{c}}{g\left( x \right) \ge 0,\;\;\;\,\;\;\;\;\;\;}\\{\left[ {\begin{array}{*{20}{c}}{f\left( x \right) = g\left( x \right),\;\;}\\{f\left( x \right) = -g\left( x \right).}\end{array}} \right.}\end{array}} \right.\)
\(\left| {{x^2} + 4x + 2} \right| = x + 2\;\;\;\; \Leftrightarrow \;\;\;\;\left\{ {\begin{array}{*{20}{c}}{x + 2 \ge 0,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,\;\;}\\{\left[ {\begin{array}{*{20}{c}}{{x^2} + 4x + 2 = x + 2,\;}\\{{x^2} + 4x + 2 = -x-2}\end{array}} \right.}\end{array}} \right.\;\;\;\; \Leftrightarrow \;\;\;\;\left\{ {\begin{array}{*{20}{c}}{x \ge -2,\;\;\;\;\;\;\;\;\;\;\;\,\;}\\{\left[ {\begin{array}{*{20}{c}}{{x^2} + 3x = 0,\;\;\;\;\,}\\{{x^2} + 5x + 4 = 0}\end{array}} \right.}\end{array}} \right.\;\;\;\; \Leftrightarrow \)
\( \Leftrightarrow \;\;\;\;\left\{ {\begin{array}{*{20}{c}}{x \ge -2,\;\;\;\;\;\;\,\;\;\;\;\;\,\,\;\,\;\;}\\{\left[ {\begin{array}{*{20}{c}}{x\left( {x + 3} \right) = 0,\;\;\;\;\;\;\,}\\{\left( {x + 4} \right)\left( {x + 1} \right) = 0}\end{array}} \right.}\end{array}} \right.\;\;\;\; \Leftrightarrow \;\;\;\;\left\{ {\begin{array}{*{20}{c}}{x \ge -2,\;\,}\\{\left[ {\begin{array}{*{20}{c}}{x = 0,\;\;}\\{x = -3,}\\{x = -4,}\\{x = -1\;}\end{array}} \right.}\end{array}} \right.\;\;\;\; \Leftrightarrow \;\;\;\;\left[ {\begin{array}{*{20}{c}}{x = -1,}\\{x = 0.\;\;}\end{array}} \right.\)
Ответ: \(-1;\;\;\;0.\)