If $a,b,c$  are positive real numbers, then the roots of the equation $a{x}^2+bx+c=0$ are

The given equation is $a{x}^2+bx+c=0$

Roots of $a{x}^2+bx+c=0$  are given by $x=-\frac{b}{2a}\pm \frac{\sqrt{b^2-4ac}}{2a}$

Since $a,b>0$ , $\therefore$ $-\frac{b}{2a}<0$

if $b^2-4ac$  is $-\text{ve}$ , then roots are imaginary having real part negative

If $b^2-4ac$  is$+\text{ve}$ , then roots real and negative.

Hence both roots have negative real parts.