Noting that $18^\circ + 12^\circ=30^\circ$ which is a special angle reminds the sum of tangent formula. \begin{align*} & \sqrt{3}\tan 18^\circ + \tan 18^\circ\tan 12^\circ +\sqrt{3}\tan 12^\circ\\ =\quad & \sqrt{3}\cdot(\tan 18^\circ + \tan 12^\circ) + \tan 18^\circ \tan 12^\circ\\ =\quad&\sqrt{3}\cdot\tan(18^\circ + 12^\circ)(1-\tan 18^\circ\tan 12^\circ) + \tan 18^\circ\tan 12^\circ\\ =\quad&\sqrt{3}\cdot\frac{\sqrt{3}}{3}\cdot(1-\tan 18^\circ\tan 12^\circ)+\tan 18^\circ\tan 12^\circ\\ =\quad& 1 \end{align*}