To solve for all possible polar coordinates for the point $(-7\sqrt{3}, -7)$, let's break down the process.

1. Convert Rectangular to Polar Coordinates: Given $(x, y) = (-7\sqrt{3}, -7)$,

   • First, find the radius $r$: $r = \sqrt{x^2 + y^2} = \sqrt{(-7\sqrt{3})^2 + (-7)^2} = \sqrt{147 + 49} = \sqrt{196} = 14$

2. Find the Angle $\theta$:

   • Using $\tan \theta = \frac{y}{x}$, $\tan \theta = \frac{-7}{-7\sqrt{3}} = \frac{1}{\sqrt{3}} = \tan\left(\frac{\pi}{6}\right)$
   • However, since both $x$ and $y$ are negative, the point is in the third quadrant, meaning we add $\pi$ to the reference angle: $\theta = \pi + \frac{\pi}{6} = \frac{7\pi}{6}$

3. Generate Polar Coordinate Pairs:

   • The polar coordinates for this point can be written as $(r, \theta + 2n\pi)$ and $(-r, \theta + \pi + 2n\pi)$ for integer $n$.
   • Thus, the polar coordinates are: $(14, \frac{7\pi}{6} + 2n\pi) \quad \text{and} \quad (-14, \frac{\pi}{6} + 2n\pi)$

Now, let's compare this with the provided answer choices. The correct answer that matches these coordinates is:

[ \boxed{\left[14, \frac{7\pi}{6} + 2n\pi\right], \left[-14, \frac{\pi}{6} + 2n\pi\right]} ]

Would you like more details on this process, or do you have further questions?

Here are some related questions you might find helpful:

1. How are polar coordinates related to rectangular coordinates?
2. Why is it necessary to add $\pi$ when finding angles in different quadrants?
3. How do you interpret negative radius values in polar coordinates?
4. What is the significance of adding $2n\pi$ in polar coordinates?
5. How do we determine the quadrant for a point based on its coordinates?

Tip: When converting rectangular coordinates to polar, always consider the signs of $x$ and $y$ to find the correct angle quadrant.