To determine the value of $a$ for which $g(x) = 0$ when $x = 0, -2,$ and $8$, we begin by evaluating $g(x)$ at these points.

The function $g$ is given by: $g(x) = x^3 + (2 - a)x^2 - 2ax$

First, evaluate $g(x)$ at $x = 0$: $g(0) = 0^3 + (2 - a)0^2 - 2a \cdot 0 = 0$

Since $g(0) = 0$, it does not provide any additional information about $a$.

Next, evaluate $g(x)$ at $x = -2$: $g(-2) = (-2)^3 + (2 - a)(-2)^2 - 2a(-2)$ $g(-2) = -8 + (2 - a)4 + 4a$ $g(-2) = -8 + 8 - 4a + 4a$ $g(-2) = -8 + 8 = 0$

This simplifies to $0 = 0$, giving no new information about $a$.

Finally, evaluate $g(x)$ at $x = 8$: $g(8) = 8^3 + (2 - a)8^2 - 2a \cdot 8$ $g(8) = 512 + (2 - a)64 - 16a$ $g(8) = 512 + 128 - 64a - 16a$ $g(8) = 512 + 128 - 80a$ $g(8) = 640 - 80a$

Since $g(8) = 0$: $640 - 80a = 0$ $80a = 640$ $a = \frac{640}{80}$ $a = 8$

Thus, the value of $a$ is $8$.

Would you like more details or have any questions? Here are 5 questions you might ask next:

1. How do you determine the roots of a polynomial equation?
2. Can you explain how to factorize cubic polynomials?
3. What are the properties of polynomial functions?
4. How can the value of $a$ affect the shape of the graph of $g(x)$?
5. What methods can be used to verify the roots of a polynomial equation?

Tip: Always check the given conditions in a problem to see if they provide independent equations, which can simplify solving for unknowns.