The question is asking which expression represents a sum of cubes. To determine this, we need to identify whether the terms in each expression can be rewritten as cubes of binomials or monomials.

Sum of cubes form:

The sum of cubes follows this pattern: $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$ Each term must be a perfect cube.

Let's analyze each option:

1. Option 1: $-27a^3b^6 + 8a^9b^{12}$

   • $-27a^3b^6 = (-3ab^2)^3$
   • $8a^9b^{12} = (2a^3b^4)^3$

   Both terms are cubes, but this is not a sum (due to the negative sign).

2. Option 2: $-9a^3b^6 + a^9b^{10}$

   • $-9a^3b^6$ and $a^9b^{10}$ are not cubes of any expression.

3. Option 3: $9a^3b^6 + 8a^9b^{12}$

   • $9a^3b^6$ is not a perfect cube.

4. Option 4: $27a^3b^6 + 8a^9b^{12}$

   • $27a^3b^6 = (3ab^2)^3$
   • $8a^9b^{12} = (2a^3b^4)^3$

   Both terms are perfect cubes, and this is a sum. Therefore, Option 4 is the correct answer.

Conclusion:

The correct expression is Option 4: $27a^3b^6 + 8a^9b^{12}$.

Would you like more details, or do you have any questions about this? Here are five related questions to consider:

1. How can you identify if a term is a perfect cube?
2. What is the difference between a sum of cubes and a difference of cubes?
3. Can you factor the sum of cubes into binomials and trinomials?
4. Why is Option 1 incorrect, despite both terms being cubes?
5. What are some real-world applications of the sum of cubes?

Tip: When checking for cubes, always look for variables and coefficients raised to powers that are multiples of 3.