If $\alpha$  and  $\beta (\alpha <\beta )$  are the roots of the equation  $x^2+bx+c=0$, where  $c<0<b$, then

Here$D=b^2-4c>0$   because $c<0<b$ . So, roots are real and unequal. Now,  $\alpha +\beta =-b<0$ and  $\alpha \beta =c<0$
Therefore, one root is positive and the other root is negative, the negative root being numerically bigger. As  $\alpha <\beta$, so $\alpha$  is the negative root while  $\beta$ is the positive root. So,  $\left|\alpha \right|>\beta$ and  $\alpha <0<\beta <\left|\alpha \right|$.