$\because$   Roots of equation   2x2 + 3x + 7 = 0  are imaginary
$\therefore$     Both the roots will be in common
$\therefore \frac{\mathrm{a}}{2}=\frac{2\mathrm{b}}{3}=\frac{3\mathrm{c}}{7}=\mathrm{k} (\mathrm{k}\in \mathrm{R})$
⇒     a = 2k,  b = $\frac{3\mathrm{k}}{2}$ and   c = $\frac{7\mathrm{k}}{3}$
For  a, b and  c  to be natural number minimum value of  k = 6
$\therefore$   a = 12,  b = 9  and  c = 14
$\therefore$   a + b + c = 35