If the equations $x^2+2x+3=0\text{ and }a{x}^2+bx+c=0,a,b,c\in R,$have a common root, then a :b : c is

$x^2+2x+3=0.....\left(1\right)$

$a{x}^2+bx+c=0.....\left(2\right)$

Since equation (i) has imaginary roots

So equation (ii) will also have both roots same as (i).Thus

$\frac{a}{1}=\frac{b}{2}=\frac{c}{3}\Rightarrow a=\lambda ,b=2\lambda ,c=3\lambda$

Hence a:b:c= 1:2:3