Simplify
\[4 \sin x \sin (60^\circ - x) \sin (60^\circ + x).\]The answer will be a trigonometric function of some simple function of $x,$ like "$\cos (2x)$" or "$\sin (x^3)$".
By product-to-sum,
\begin{align*}
4 \sin x \sin (60^\circ - x) \sin (60^\circ + x) &= 4 \sin x \cdot \frac{1}{2} (\cos 2x - \cos 120^\circ) \\
&= 2 \sin x \left( \cos 2x + \frac{1}{2} \right) \\
&= 2 \sin x \cos 2x + \sin x.
\end{align*}Again by product-to-sum,
\begin{align*}
2 \sin x \cos 2x + \sin x &= \sin 3x + \sin (-x) + \sin x \\
&= \boxed{\sin 3x}.
\end{align*}