The expression $8b^3 + 27c^3$ is a sum of cubes. This can be factored using the sum of cubes formula, which is:

$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$

In this case, we can identify:

$a = 2b \quad \text{and} \quad b = 3c$

Applying the sum of cubes formula:

$8b^3 + 27c^3 = (2b)^3 + (3c)^3 = (2b + 3c)\left((2b)^2 - (2b)(3c) + (3c)^2\right)$

Now, let's expand and simplify the expression inside the parentheses:

1. $(2b)^2 = 4b^2$
2. $(2b)(3c) = 6bc$
3. $(3c)^2 = 9c^2$

So the factored form of the expression is:

$8b^3 + 27c^3 = (2b + 3c)(4b^2 - 6bc + 9c^2)$

Would you like more details, or do you have any questions?

Related Questions:

1. How do you factor a difference of cubes?
2. What is the general formula for factoring a sum of cubes?
3. How can the sum of cubes be applied in solving polynomial equations?
4. What are other common factoring techniques for polynomials?
5. How do you verify the factorization of a polynomial?
6. Can the sum of cubes formula be extended to higher powers?
7. How would you factor $8b^3 - 27c^3$ (a difference of cubes)?
8. What are some practical applications of the sum of cubes in algebra?

Tip:

When factoring sums or differences of cubes, always double-check by expanding the factors to ensure accuracy.