[tex]x^2+mx+2m+8 = 0; \quad x_1 = 2x_2 \\ \\ x_1+x_2 = -\dfrac{b}{a} = -m \\ x_1x_2 = \dfrac{c}{a} =2m+8 \\ \\ x_1 = 2x_2 \Rightarrow x_1-2x_2 = 0 \Rightarrow x_1+x_2 - 3x_2 = 0 \Rightarrow -m-3x_2 = 0 \Rightarrow \\ \Rightarrow -3x_2 = m \Rightarrow \boxed{x_2 = -\dfrac{m}{3}}[/tex]
[tex]
$ Acum trebuie neaparat sa il inlocuim pe x_2 $ relatia $ x_1\cdot x_2 =2m+8, \\ $ deoarece daca il inlocuim in relatia $ x_1+x_2 = -m , $ adica, relatia cu $ \\$ajutorul careia l-am scos, nu vom rezolva nimic, se va reduce m cu m. \\ \\ x_1\cdot x_2 = 2m+8 \Rightarrow x_1\cdot\Big(-\dfrac{m}{3}\Big) = 2m+8 \Rightarrow x_1 = \dfrac{2m+8}{-\dfrac{m}{3}} \Rightarrow \\ \\ \Rightarrow x_1 = - \dfrac{3\cdpt(2m+8)}{m} \Rightarrow \boxed{x_1 =- \dfrac{6m+24}{m}} [/tex]
[tex]$ \ $Acum inlocuim solutiile in relatia: $x_1 = 2x_2 \\ \\ \Rightarrow -\dfrac{6m+24}{m}=-2\cdot\dfrac{m}{3} \Rightarrow \dfrac{6m+24}{m} = 2\cdot \dfrac{m}{3} \Rightarrow 18m+72 = 2m^2 \Rightarrow \\ \\ \Rightarrow 2m^2-18m-72 = 0 \Rightarrow m^2 -9m-36 = 0 \\ \\ \Delta = (-9)^2+-4\cdot1\cdot(-36) = 81+144 = 225 = 15^2 \\ \\ \Rightarrow m_{1,2} = \dfrac{9\pm15}{2} \Rightarrow \left\| \begin{array}{c} m_1 = -3 \\ m_2 = 12 \end{array} \right \Rightarrow \boxed{\boxed{m\in\Big\{\Big -3;12\Big\}}}[/tex]
