We are asked to evaluate the function $h(x) = -3x^4 + 2x^3 - 12x - 6$ for $x = -2$.

To do this, substitute $x = -2$ into the function:

$h(-2) = -3(-2)^4 + 2(-2)^3 - 12(-2) - 6$

Now, we will calculate each term:

1. $(-2)^4 = 16$, so $-3(-2)^4 = -3 \times 16 = -48$
2. $(-2)^3 = -8$, so $2(-2)^3 = 2 \times (-8) = -16$
3. $-12(-2) = 24$
4. The constant term is just $-6$.

Now, substitute these values back into the equation:

$h(-2) = -48 - 16 + 24 - 6$

Now, perform the arithmetic:

$h(-2) = -48 - 16 = -64$ $-64 + 24 = -40$ $-40 - 6 = -46$

Thus, the value of $h(-2)$ is $\boxed{-46}$.

Do you need further explanation or details?

Here are 5 related questions for you:

1. What is the general approach to evaluating polynomial functions?
2. How would the function change if we had a different value of $x$, say $x = 3$?
3. How can we use synthetic division to evaluate a polynomial at a given value of $x$?
4. What is the significance of higher-degree terms in a polynomial function?
5. How do you find the derivative of a polynomial function?

Tip: When evaluating polynomials, always carefully apply exponent rules and perform arithmetic step-by-step to avoid mistakes.