Solve {l}{2x^2+(2m+1)x+2m+1=0}{x_1^2+x_2^2>1}

$$\left\{ \begin{array} { l } { 2 x ^ { 2 } + ( 2 m + 1 ) x + 2 m + 1 = 0 } \\ { x _ { 1 } ^ { 2 } + x _ { 2 } ^ { 2 } > 1 } \end{array} \right.$$

Solve for x

$$\left\{\begin{matrix}x=-\frac{\sqrt{4m^{2}-12m-7}}{4}-\frac{m}{2}-\frac{1}{4}\text{; }x=\frac{\sqrt{4m^{2}-12m-7}}{4}-\frac{m}{2}-\frac{1}{4}\text{, }&\left(m\leq -\frac{1}{2}\text{ and }|x_{1}|>\sqrt{1-x_{2}^{2}}\text{ and }|x_{2}|\leq 1\right)\text{ or }\left(|x_{2}|>1\text{ and }m\leq -\frac{1}{2}\right)\\x=-\frac{\sqrt{4m^{2}-12m-7}}{4}-\frac{m}{2}-\frac{1}{4}\text{; }x=\frac{\sqrt{4m^{2}-12m-7}}{4}-\frac{m}{2}-\frac{1}{4}\text{, }&\left(m\geq \frac{7}{2}\text{ and }|x_{1}|>\sqrt{1-x_{2}^{2}}\text{ and }|x_{2}|\leq 1\right)\text{ or }\left(m\geq \frac{7}{2}\text{ and }|x_{2}|>1\right)\end{matrix}\right.$$

Solve for x_2

$$\left\{\begin{matrix}x_{2}\in \mathrm{R}\text{, }&|x_{1}|>1\text{ and }m=-\frac{2x^{2}+x+1}{2\left(x+1\right)}\text{ and }x\neq -1\\x_{2}\in \left(\sqrt{1-x_{1}^{2}},\infty\right)\cup \left(-\infty,-\sqrt{1-x_{1}^{2}}\right)\text{, }&m=-\frac{2x^{2}+x+1}{2\left(x+1\right)}\text{ and }x\neq -1\end{matrix}\right.$$