If the equations $a{x}^2+bx+c=0$ and $2x^2+3x+4=0$a common root, then $a : b : c$

The equation $2x^2+3x+4=0$ has complex roots

which always occur in pairs. So, the two equations have both roots common.

$\therefore \frac{a}{2}=\frac{b}{3}=\frac{c}{4}\Rightarrow a : b : c=2 : 3 : 4$