A $120^\circ$ rotation around the origin in the clockwise direction is applied to $4 - \sqrt{3} + (-1 - 4 \sqrt{3})i.$  What is the resulting complex number?
A $120^\circ$ rotation around the origin in the clockwise direction corresponds to multiplication by $\operatorname{cis} (-120)^\circ = -\frac{1}{2} + \frac{\sqrt{3}}{2} i.$

[asy]
unitsize(0.5 cm);

pair A = (4 - sqrt(3),-1 - 4*sqrt(3)), B = (-8,2);

draw((-8,0)--(8,0));
draw((0,-8)--(0,3));
draw((0,0)--A,dashed);
draw((0,0)--B,dashed);

dot("$4 - \sqrt{3} + (-1 - 4 \sqrt{3})i$", A, S);
dot("$-8 + 2i$", B, W);
[/asy]

Thus, the image of $4 - \sqrt{3} + (-1 - 4 \sqrt{3})i$ is
\[(4 - \sqrt{3} + (-1 - 4 \sqrt{3})i) \left( -\frac{1}{2} - \frac{\sqrt{3}}{2} i \right) = \boxed{-8 + 2i}.\]