1. Let $x=\cos\theta$. Then the equation becomes

$2\sqrt{2}x^2+(2-\sqrt{6})x-\sqrt{3}=0.$

We must first solve this quadratic for $x$, and then count how many $\theta\in[-2\pi,2\pi]$ satisfy $\cos\theta=x$.

2. Factor the quadratic.

We look for a factorization of the form

$2\sqrt{2}x^2+(2-\sqrt{6})x-\sqrt{3}=(ax+b)(cx+d).$

Try

$(\sqrt{2}x+\sqrt{3})(2x-1).$

Expanding:

$$(\sqrt{2}x+\sqrt{3})(2x-1) =2\sqrt{2}x^2-\sqrt{2}x+2\sqrt{3}x-\sqrt{3}.$$

Now,

$-\sqrt{2}+2\sqrt{3}=2-\sqrt{6}?$

Let us instead try

$(\sqrt{2}x-\sqrt{3})(2x+1).$

Expanding:

$$(\sqrt{2}x-\sqrt{3})(2x+1) =2\sqrt{2}x^2+\sqrt{2}x-2\sqrt{3}x-\sqrt{3}$$

so the coefficient of $x$ is

$\sqrt{2}-2\sqrt{3}.$

This is not equal to $2-\sqrt{6}$ directly. So we solve using the quadratic formula.

3. Use the quadratic formula for

$2\sqrt{2}x^2+(2-\sqrt{6})x-\sqrt{3}=0.$

Here,

$a=2\sqrt{2},\quad b=2-\sqrt{6},\quad c=-\sqrt{3}.$

Then

$x=\frac{-(2-\sqrt{6})\pm\sqrt{(2-\sqrt{6})^2-4(2\sqrt{2})(-\sqrt{3})}}{2\cdot 2\sqrt{2}}.$

Compute the discriminant:

$$(2-\sqrt{6})^2+8\sqrt{6} =4+6-4\sqrt{6}+8\sqrt{6} =10+4\sqrt{6}.$$

Notice that

$10+4\sqrt{6}=(2+\sqrt{6})^2.$

So

$\sqrt{10+4\sqrt{6}}=2+\sqrt{6}.$

Hence

$x=\frac{-2+\sqrt{6}\pm(2+\sqrt{6})}{4\sqrt{2}}.$

Thus the two roots are:

• With $+$ sign: $x=\frac{-2+\sqrt{6}+2+\sqrt{6}}{4\sqrt{2}}=\frac{2\sqrt{6}}{4\sqrt{2}}=\frac{\sqrt{3}}{2}.$

• With $-$ sign: $x=\frac{-2+\sqrt{6}-2-\sqrt{6}}{4\sqrt{2}}=\frac{-4}{4\sqrt{2}}=-\frac{1}{\sqrt{2}}=-\frac{\sqrt{2}}{2}.$

So the equation reduces to

$\cos\theta=\frac{\sqrt{3}}{2} \quad \text{or} \quad \cos\theta=-\frac{\sqrt{2}}{2}.$

4. Count solutions for $\cos\theta=\frac{\sqrt{3}}{2}$ in $[-2\pi,2\pi]$.

General angles are

$\theta=2k\pi\pm\frac{\pi}{6}.$

List those in $[-2\pi,2\pi]$:

$-\frac{11\pi}{6},\ -\frac{\pi}{6},\ \frac{\pi}{6},\ \frac{11\pi}{6}.$

So there are 4 solutions.

5. Count solutions for $\cos\theta=-\frac{\sqrt{2}}{2}$ in $[-2\pi,2\pi]$.

General angles are

$\theta=2k\pi\pm\frac{3\pi}{4}.$

List those in $[-2\pi,2\pi]$:

$-\frac{5\pi}{4},\ -\frac{3\pi}{4},\ \frac{3\pi}{4},\ \frac{5\pi}{4}.$

So there are 4 solutions.

6. Total number of solutions:

$4+4=8.$

Therefore, the correct option is

$\boxed{\text{A: }8}.$

7. Comparison with stored answer:

Stored correct answer is A, which matches our derived answer.