10. For the equation $x^{2}+m x+1+2 \mathrm{i}=0$ with respect to $x$, if it has real roots, then the minimum value of the modulus of the complex number $m$ is
10. $\sqrt{2+2 \sqrt{5}}$ Let $m=x+y i, x, y \in \mathbf{R}, \alpha$ be a real root of the equation, $\alpha \neq 0$, and $\alpha^{2}+a x+1=0, 2 y+2=$ $0 .|m|^{2}=x^{2}+y^{2}=\left(-\alpha-\frac{1}{\alpha}\right)^{2}+\left(-\frac{2}{\alpha}\right)^{2}=2+\alpha^{2}+\frac{5}{\alpha^{2}} \geqslant 2+2 \sqrt{5}, \alpha=\sqrt[4]{5}$ when the equality holds.