\[\begin{array}{l}
4\sin x.\sin 2x.\sin 3x\\
= 4.\sin x.\dfrac{1}{2}\left[ {\cos \left( {2x - 3x} \right) - \cos \left( {2x + 3x} \right)} \right]\\
= 2\sin x\left( {\cos x - \cos 5x} \right)\\
= 2\sin x\cos x - 2\sin x.\cos 5x\\
= \sin 2x - \left[ {\sin 6x + \sin \left( { - 4x} \right)} \right]\\
= \sin 2x + \sin 4x - \sin 6x
\end{array}\]