$\left(3-m\right)x^2+\left(m+1\right)x-{\left(m+1\right)}^2=0$
$\left(3\right)x_1^2+x_2^2=x_1\cdot x_2+7$
        $x_1^2+2x_1{x}_2+x_2^2-2x_1{x}_2=x_1\cdot x_2+7$
        ${\left(x_1+x_2\right)}^2-3x_1{x}_2=7$
        ${\left(\frac{-\left(m+1\right)}{\left(3-m\right)}\right)}^2-3\cdot \frac{-{\left(m+1\right)}^2}{3-m}=7$
        ${\left(m+1\right)}^2+3{\left(m+1\right)}^2\left(3-m\right)=7{\left(3-m\right)}^2$
        ${\left(m+1\right)}^2\left(1+3\left(3-m\right)\right)=7{\left(3-m\right)}^2$
        ${\left(m+1\right)}^2\left(10-3m\right)=7{\left(3-m\right)}^2$
        $\left(m^2+2m+1\right)\left(10-3m\right)=7{\left(3-m\right)}^2$
        $10m^2+20m+10-3m^3-6m^2-3m=7{\left(3-m\right)}^2$
        $-3m^3+4m^2+17m+10=63-42m+7m^2$
        $-3m^3-3m^2+59m-53=0$
        $3m^3+3m^2-59m+53=0$

$\left(m-1\right)\left(3m^2+6m-53\right)=0$
$m=1\vee 3m^2+6m-53=0$

$3m^2+6m-53=0$
${\Delta }_m=6^2-4\cdot 3\cdot \left(-53\right)=672$
$\sqrt{{\Delta }_m}=4\sqrt{42}$
$m_1=\frac{-6-4\sqrt{42}}{6}=\frac{-3-2\sqrt{42}}{3}$
$m_2=\frac{-6+4\sqrt{42}}{6}=\frac{-3+2\sqrt{42}}{3}$