Let $P(z) = z^8 + (4\sqrt{3} + 6) z^4 - (4\sqrt{3}+7)$. What is the minimum perimeter among all the 8-sided polygons in the complex plane whose vertices are precisely the zeros of $P(z)$?

$ \textbf{(A)}\ 4\sqrt{3}+4  \qquad
\textbf{(B)}\ 8\sqrt{2}  \qquad
\textbf{(C)}\ 3\sqrt{2}+3\sqrt{6}  \qquad
\textbf{(D)}\ 4\sqrt{2}+4\sqrt{3}  \qquad
$
$\textbf{(E)}\ 4\sqrt{3}+6  $
1. **Factor the polynomial**: We start by factoring the polynomial \( P(z) = z^8 + (4\sqrt{3} + 6) z^4 - (4\sqrt{3} + 7) \). Notice that it can be factored as:
   \[
   P(z) = (z^4 - 1)(z^4 + 4\sqrt{3} + 7)
   \]
   This factorization gives us the roots of \( P(z) \).

2. **Find the roots**: The roots of \( P(z) \) are the solutions to \( z^4 - 1 = 0 \) and \( z^4 + 4\sqrt{3} + 7 = 0 \).
   - The roots of \( z^4 - 1 = 0 \) are \( \pm 1 \) and \( \pm i \).
   - For \( z^4 + 4\sqrt{3} + 7 = 0 \), we solve for \( z \):
     \[
     z^4 = -(4\sqrt{3} + 7)
     \]
     Notice that \( (2 + \sqrt{3})^2 = 4 + 4\sqrt{3} + 3 = 4\sqrt{3} + 7 \), so:
     \[
     z^4 = -(2 + \sqrt{3})^2
     \]
     The roots are \( (2 + \sqrt{3})^{1/4} e^{i(\pi/4 + k\pi/2)} \) for \( k = 0, 1, 2, 3 \).

3. **Magnitude of the roots**: The magnitude of the roots \( (2 + \sqrt{3})^{1/4} \) can be simplified:
   \[
   (2 + \sqrt{3})^{1/4} = \sqrt[4]{2 + \sqrt{3}}
   \]

4. **Vertices of the polygon**: The vertices of the 8-sided polygon are \( \pm 1, \pm i \), and the four roots \( \sqrt[4]{2 + \sqrt{3}} e^{i(\pi/4 + k\pi/2)} \).

5. **Calculate the distance between vertices**: The vertices form a symmetric shape. We need to find the distance between \( 1 \) and \( \sqrt[4]{2 + \sqrt{3}} e^{i\pi/4} \):
   \[
   \sqrt[4]{2 + \sqrt{3}} e^{i\pi/4} = \sqrt[4]{2 + \sqrt{3}} \left( \frac{\sqrt{2}}{2} + i \frac{\sqrt{2}}{2} \right)
   \]
   The distance \( d \) between \( 1 \) and this root is:
   \[
   d = \sqrt{\left(1 - \sqrt[4]{2 + \sqrt{3}} \frac{\sqrt{2}}{2}\right)^2 + \left(\sqrt[4]{2 + \sqrt{3}} \frac{\sqrt{2}}{2}\right)^2}
   \]
   Simplifying the expression inside the square root:
   \[
   d = \sqrt{\left(1 - \sqrt[4]{2 + \sqrt{3}} \frac{\sqrt{2}}{2}\right)^2 + \left(\sqrt[4]{2 + \sqrt{3}} \frac{\sqrt{2}}{2}\right)^2}
   \]
   Recognizing that \( \sqrt[4]{2 + \sqrt{3}} = \sqrt{1 + \frac{\sqrt{3}}{2}} \):
   \[
   d = \sqrt{3 + \sqrt{3} - \sqrt{4 + 2\sqrt{3}}}
   \]
   Since \( (1 + \sqrt{3})^2 = 4 + 2\sqrt{3} \):
   \[
   d = \sqrt{3 + \sqrt{3} - 1 - \sqrt{3}} = \sqrt{2}
   \]

6. **Calculate the perimeter**: Since there are 8 sides of this length, the total perimeter of the polygon is:
   \[
   8 \times \sqrt{2} = 8\sqrt{2}
   \]

The final answer is \(\boxed{8\sqrt{2}}\)