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

For $x^2+2x+3=0$, discriminant $D=2^2-4(1)(3)=4-12=-8<0$.The roots are imaginary. Since coefficients $a,b,c$ are real, if one root is common, the other conjugate root must also be common.This means the two equations are identical (roots are the same).
$$\frac{a}{1}=\frac{b}{2}=\frac{c}{3}$$
Therefore, $a:b:c=1:2:3$.