If the equation (a + 2)x² + bx + c = 0 and 2x² + 3x + 4 = 0 have a common root where a, b, c $\in$ N then

D < 0 for 

(a + 2)x² + bx + c = 0   ...(i)

2x² + 3x + 4 = 0    ...(ii)
$\Rightarrow$ (i) and (ii) have both roots common.
for equation (i).
$\begin{array}{l}(\mathrm{a}+2){\mathrm{x}}^2+\mathrm{bx}+\mathrm{c}=0\\ \mathrm{D}<0\\ \Rightarrow {\mathrm{b}}^2-4\mathrm{c}(\mathrm{a}+2)<0\Rightarrow {\mathrm{b}}^2<4\mathrm{ac}+8\mathrm{c}\\ \Rightarrow \frac{\mathrm{a}+2}{2}=\frac{\mathrm{b}}{3}=\frac{\mathrm{c}}{4}=\mathrm{k}\left(\text{ let }\right)\\ \Rightarrow \mathrm{k}\in \mathrm{N},\mathrm{k}\ge 2\frac{\mathrm{a}+\mathrm{b}+\mathrm{c}+2}{9}=\mathrm{k}\\ \Rightarrow \mathrm{a}+\mathrm{b}+\mathrm{c}=9\mathrm{k}-2(\mathrm{a}+\mathrm{b}+\mathrm{c}{)}_{min}=16\end{array}$