Sum of the real roots of the equation $\begin{vmatrix} 1 & 4 & 20\\ 1 & -2 & 5 \\ 1 & 2x & 5{x}^{2} \end{vmatrix}=0$ is

$\begin{array}{l} \\ \left|\begin{array}{lll} 1 & 4 & 20 \\ 1 & -2 & 5 \\ 1 & 2 x & 5 x^{2} \end{array}\right|=0 \\ \Rightarrow 1\left|\begin{array}{cc} -2 & 5 \\ 2 x & 5 x^{2} \end{array}\right|-4\left|\begin{array}{cc} 1 & 5 \\ 1 & 5 x^{2} \end{array}\right|+20\left|\begin{array}{cc} 1 & -2 \\ 1 & 2 x \end{array}\right|=0 \\ \Rightarrow 1\left(-10 x^{2}-10 x\right)-4\left(5 x^{2}-5\right)+20(2 x+2)=0 \\ \Rightarrow-10 x^{2}-10 x-20 x^{2}+20+40 x+40=0 \\ \Rightarrow-30 x^{2}+30 x+60=0 \\ \Rightarrow-x^{2}+x+2=0 \quad\left(\frac{1}{1}-30\right) \\ \Rightarrow x^{2}-x-2=0 \quad(x-1) \\ \Rightarrow x^{2}-2 x+x-2=0 \Rightarrow x(x-2)+1(x-2)=0 \\ \Rightarrow x=-1,2 \varepsilon \mathbb{R} \Rightarrow \operatorname{sum}=-1+2=1 . \end{array}$