To solve the given equation and determine the value of $k$, we start by analyzing the expression:

$\frac{5}{8}(5x + 8)(x + \sqrt{5k} + 8)(x - \sqrt{5k} + 8) = 0$

This equation will be zero if any one of the factors is zero. Therefore, we need to solve for $x$ in the following individual equations:

1. $5x + 8 = 0$
2. $x + \sqrt{5k} + 8 = 0$
3. $x - \sqrt{5k} + 8 = 0$

First, solve $5x + 8 = 0$:

$5x + 8 = 0$ $5x = -8$ $x = -\frac{8}{5}$

Next, solve $x + \sqrt{5k} + 8 = 0$:

$x + \sqrt{5k} + 8 = 0$ $x = -\sqrt{5k} - 8$

And solve $x - \sqrt{5k} + 8 = 0$:

$x - \sqrt{5k} + 8 = 0$ $x = \sqrt{5k} - 8$

We now have three solutions for $x$:

1. $x_1 = -\frac{8}{5}$
2. $x_2 = -\sqrt{5k} - 8$
3. $x_3 = \sqrt{5k} - 8$

The product of the solutions $x_1$, $x_2$, and $x_3$ is given as 64. We write the product of the roots:

$x_1 \cdot x_2 \cdot x_3 = 64$

Substitute the values of $x_1$, $x_2$, and $x_3$:

$\left( -\frac{8}{5} \right) \left( -\sqrt{5k} - 8 \right) \left( \sqrt{5k} - 8 \right) = 64$

Simplify the product:

$\left( -\frac{8}{5} \right) \left( (\sqrt{5k})^2 - 8^2 \right) = 64$ $\left( -\frac{8}{5} \right) \left( 5k - 64 \right) = 64$

Next, distribute $-\frac{8}{5}$:

$-\frac{8}{5} \cdot (5k - 64) = 64$ $-8k + \frac{8 \cdot 64}{5} = 64$ $-8k + \frac{512}{5} = 64$

Multiply everything by 5 to clear the fraction:

$-40k + 512 = 320$ $-40k = 320 - 512$ $-40k = -192$ $k = \frac{192}{40}$ $k = 4.8$

Thus, the value of $k$ is $4.8$.

Would you like more details or have any questions? Here are 5 questions you might find interesting:

1. How do you factor quadratic equations?
2. What are the properties of polynomial roots?
3. How do you solve systems of equations?
4. What is the quadratic formula, and how is it derived?
5. How do you handle complex roots in polynomials?

Tip: When solving polynomial equations, always consider factoring or using the quadratic formula if the polynomial is quadratic.