Cho các tập hợp: \(A = \left\{ {\left. {x \in \mathbb{R}} \right|\,\left( {{x^2} + 7x + 6} \right)\left( {{x^2} - 4} \right) = 0} \right\};\) \(B = \left\{ {x \in \mathbb{N}|\,2x \le 8} \right\}\);   \(C = {\rm{\{ }}2x + 1|\,x \in \mathbb{Z};\,\,\, - 2 \le x \le 4{\rm{.}}\) Tìm  \(\left( {A \cup C} \right)\backslash B.\)

\( \bullet \) Ta có: \(\left( {{x^2} + 7x + 6} \right)\left( {{x^2} - 4} \right) = 0 \Leftrightarrow \left[ \begin{array}{l}{x^2} + 7x + 6 = 0\\{x^2} - 4 = 0\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}x =  - 1\\x =  - 6\\x =  - 2\\x = 2\end{array} \right..\)

 Vậy \(A = \left\{ { - 6; - 2; - 1;2} \right\}.\)

\( \bullet \) Ta có: \(\left\{ \begin{array}{l}x \in \mathbb{N}\\2x \le 8\end{array} \right. \)\(\Leftrightarrow \left\{ \begin{array}{l}x \in \mathbb{N}\\x \le 4\end{array} \right. \)\(\Leftrightarrow x \in \left\{ {0;\,\,1;\,\,2;\,\,3;\,\,4} \right\}.\)

Vậy \(B = \left\{ {0;\,\,1;\,\,2;\,\,3;\,\,4} \right\}.\)

\( \bullet \) Ta có: \(\left\{ \begin{array}{l}x \in \mathbb{Z}\\ - 2 \le x \le 4\end{array} \right.\)\( \Leftrightarrow x \in \left\{ { - 2; - 1;\,\,0;\,\,1;\,\,2;\,\,3 ;\,\,4} \right\} \)\(\Rightarrow \left( {2x + 1} \right) \in \left\{ { - 3; - 1;\,1;\,3;\,\,5;\,\,7;\,\,9} \right\}.\)

\( \Rightarrow C = \left\{ { - 3; - 1;\,1;\,3;\,5;\,7;\,9} \right\}\)

Ta có: \(A \cup C = \left\{ { - 6; - 3; - 2; - 1;1;2;3;5;7;9} \right\}\)     

\( \Rightarrow \left( {A \cup C} \right)\backslash B = \left\{ { - 6; - 3; - 2; - 1;\,5;\,7;\,9} \right\}.\)