6. A particle travels along the $x$-axis, its velocity described by $\frac{d x}{d t}=6(1+t)(2-t)$. At what two times does the particle reverse its direction, and how far does it travel to get from one such point to the other?

Step 1 ：The particle reverses its direction when its velocity changes sign. This happens when the velocity function \(\frac{d x}{d t}=6(1+t)(2-t)\) equals zero. So, we need to solve the equation \(6(1+t)(2-t)=0\) for t.

Step 2 ：The times at which the particle reverses its direction are \(t=-1\) and \(t=2\).

Step 3 ：We can find the distance the particle travels between these two points by integrating the absolute value of the velocity function from the smaller time to the larger time.

Step 4 ：The distance the particle travels between these two points is 27 units.

Step 5 ：Final Answer: The particle reverses its direction at times \(t=-1\) and \(t=2\). The distance the particle travels between these two points is \(\boxed{27}\) units.