The function given is $g(x) = -2x^2 - 8x - 6$.

This is a quadratic function of the form $g(x) = ax^2 + bx + c$, where $a = -2$, $b = -8$, and $c = -6$.

To determine whether the function has a maximum or minimum value, we need to examine the coefficient $a$. Since $a$ is negative ($a = -2$), the parabola opens downwards, meaning the function has a maximum value.

The maximum or minimum value of a quadratic function occurs at the vertex of the parabola. The $x$-coordinate of the vertex can be found using the formula:

$x = \frac{-b}{2a}$

Substituting the values of $a$ and $b$:

$x = \frac{-(-8)}{2(-2)} = \frac{8}{-4} = -2$

Now, substitute $x = -2$ back into the function to find the corresponding $y$-value:

$g(-2) = -2(-2)^2 - 8(-2) - 6$

$g(-2) = -2(4) + 16 - 6$

$g(-2) = -8 + 16 - 6 = 2$

So, the maximum value of $g(x)$ is 2.

The correct answer is:

d. 2

Would you like more details or have any questions about this problem? Here are some related questions:

1. How do you determine the vertex of a quadratic function?
2. What does the sign of the leading coefficient indicate about the graph of a quadratic function?
3. Can a quadratic function have both a maximum and minimum value?
4. How do you complete the square to find the vertex form of a quadratic function?
5. How can you find the axis of symmetry for a quadratic function?
6. What are the effects of changing the coefficient $a$ in a quadratic function?
7. How do you determine if a quadratic equation has real roots?
8. What is the significance of the discriminant in quadratic equations?

Tip: For any quadratic function, the vertex formula $x = \frac{-b}{2a}$ quickly gives you the $x$-value at which the function reaches its maximum or minimum.