$x^2-\left(m+1\right)x+m=0$
$(1)\Delta >0\to x\in \mathbb{R}\setminus \left\{1\right\}$

$(2)x_1\ne 0\wedge x_2\ne 0\to m\in \mathbb{R}\setminus \left\{0\right\}$

$(3)\frac{1}{x_1}+\frac{1}{x_2}+2=\frac{1}{x_1^2}+\frac{1}{x_2^2}$
        $\frac{x_2}{x_1{x}_2}+\frac{x_1}{x_1{x}_2}+2=\frac{x_2^2}{x_1^2{x}_2^2}+\frac{x_1^2}{x_1^2{x}_2^2}$
        $\frac{x_2+x_1}{x_1{x}_2}+2=\frac{x_2^2+x_1^2}{x_1^2{x}_2^2}$
        $\frac{x_2+x_1}{x_1{x}_2}+2=\frac{x_2^2+2x_1{x}_2+x_1^2-2x_1{x}_2}{{\left(x_1{x}_2\right)}^2}$
        $\frac{x_1+x_2}{x_1{x}_2}+2=\frac{{\left(x_1^{}+x_2\right)}^2-2x_1{x}_2}{{\left(x_1{x}_2\right)}^2}$
        $\frac{\frac{m+1}{1_{}}}{\frac{m^{}}{1}}+2=\frac{{\left(\frac{m+1}{1_{}}\right)}^2-2\cdot \frac{m^{}}{1}}{{\left(\frac{m^{}}{1}\right)}^2}$
        $\frac{m+1}{m}+2=\frac{{\left(m+1\right)}^2-2m}{m^2}$
    

$\frac{m+1}{m}+2=\frac{m^2+2m+1-2m}{m^2}$
$\frac{m+1}{m}+2=\frac{m^2+1}{m^2}|\cdot m^2$
$m\left(m+1\right)+2m^2=m^2+1$
$m^2+m+2m^2-m^2-1=0$
$2m^2+m-1=0$
${\Delta }_m=1^2-4\cdot 2\cdot \left(-1\right)=1+8=9$
$\sqrt{{\Delta }_m}=3$
$m_1=\frac{-1-3}{2\cdot 2}=\frac{-4}{4}=-1$
$m_2=\frac{-1+3}{2\cdot 2}=\frac{2}{4}=\frac{1}{2}$

Odp.: $m\in \left\{-1,\frac{1}{2}\right\}$