The problem asks us to solve the equation:

$9^{x-8} = 3^{4x-12}.$

Step-by-step Solution:

1. Rewrite the bases: Since $9 = 3^2$, we can rewrite $9^{x-8}$ as $(3^2)^{x-8} = 3^{2(x-8)}$. Now the equation becomes: $3^{2(x-8)} = 3^{4x-12}.$

2. Simplify the exponents: Using the property $(a^m)^n = a^{mn}$, the left-hand side becomes $3^{2x - 16}$. Thus: $3^{2x - 16} = 3^{4x - 12}.$

3. Equating the exponents: Since the bases are the same, equate the exponents: $2x - 16 = 4x - 12.$

4. Solve for $x$: Simplify the equation: $2x - 4x = -12 + 16,$ $-2x = 4,$ $x = -2.$

Final Answer:

The solution is: $\boxed{\text{D. } -2}.$

Would you like further clarification or details?

Here are 5 related questions to deepen your understanding:

1. How do you simplify exponents when rewriting bases?
2. What properties of exponents are key to solving such equations?
3. Why is it necessary to equate the exponents in equations with the same base?
4. How can this type of problem change with different bases (e.g., $2^x$ vs $3^x$)?
5. Can you solve this equation graphically?

Tip: Always simplify the bases of exponential equations to a common base when possible; it simplifies the solution process significantly.