To solve \(\sin(\theta + 45^\circ) = 2 \cos(\theta - 30^\circ)\), we use the angle addition formulas:

\(\sin(\theta + 45^\circ) = \sin \theta \cos 45^\circ + \cos \theta \sin 45^\circ\)

\(\cos(\theta - 30^\circ) = \cos \theta \cos 30^\circ + \sin \theta \sin 30^\circ\)

Substitute these into the equation:

\(\sin \theta \cos 45^\circ + \cos \theta \sin 45^\circ = 2(\cos \theta \cos 30^\circ + \sin \theta \sin 30^\circ)\)

Using \(\cos 45^\circ = \sin 45^\circ = \frac{\sqrt{2}}{2}\), \(\cos 30^\circ = \frac{\sqrt{3}}{2}\), and \(\sin 30^\circ = \frac{1}{2}\), we have:

\(\frac{\sqrt{2}}{2} \sin \theta + \frac{\sqrt{2}}{2} \cos \theta = 2 \left( \frac{\sqrt{3}}{2} \cos \theta + \frac{1}{2} \sin \theta \right)\)

Simplify:

\(\frac{\sqrt{2}}{2} \sin \theta + \frac{\sqrt{2}}{2} \cos \theta = \sqrt{3} \cos \theta + \sin \theta\)

Rearrange terms:

\(\frac{\sqrt{2}}{2} \sin \theta - \sin \theta = \sqrt{3} \cos \theta - \frac{\sqrt{2}}{2} \cos \theta\)

\(\left( \frac{\sqrt{2}}{2} - 1 \right) \sin \theta = \left( \sqrt{3} - \frac{\sqrt{2}}{2} \right) \cos \theta\)

Divide both sides by \(\cos \theta\):

\(\tan \theta = \frac{\sqrt{3} - \frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2} - 1}\)

Calculate \(\theta\):

\(\theta = \tan^{-1}\left( \frac{\sqrt{3} - \frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2} - 1} \right) \approx 105.9^\circ\)

Thus, the solution is \(\theta = 105.9^\circ\).