$\left(m+3\right)x^2-\left(m+2\right)x-1=0$

$\left(1\right)m+3\ne 0\to m\ne -3$
$\left(2\right)\Delta >0\to m\ne -4$
$\left(3\right)x_1>0\wedge x_2>0\to m\in \left(-\infty ,-3\right)$

$\left(4\right)x_1^2+x_2^2>3$
        $x_1^2+x_2^2+2x_1{x}_2-2x_1{x}_2>3$
        ${\left(x_1+x_2\right)}^2-2x_1{x}_2>3$
        ${\left(\frac{m+2}{m+3}\right)}^2-2\cdot \frac{-1}{m+3}>3|\cdot {\left(m+3\right)}^2$
        ${\left(m+2\right)}^2+2\left(m+3\right)>3{\left(m+3\right)}^2$
        $m^2+4m+4+2m+6>3m^2+18m+27$
        $-2m^2-12m-17>0$
        ${\Delta }_m={\left(-12\right)}^2-4\cdot \left(-2\right)\cdot \left(-17\right)=8$
        $\sqrt{{\Delta }_m}=\sqrt{8}=\sqrt{4\cdot 2}=2\sqrt{2}$
        $m_1=\frac{12-2\sqrt{2}}{-4}=\frac{\sqrt{2}-6}{2}$
        $m_2=\frac{12+2\sqrt{2}}{-4}=\frac{-\sqrt{2}-6}{2}$

$m\in \left(\frac{-\sqrt{2}-6}{2},\frac{\sqrt{2}-6}{2}\right)$

Odp.: $m\in \left(\frac{-\sqrt{2}-6}{2},-3\right)$