The given expression $8a^3 + 125b^3$ can be factored using the sum of cubes formula:

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

Let's rewrite the expression:

$8a^3 + 125b^3$

First, recognize that $8a^3$ and $125b^3$ can be written as $(2a)^3$ and $(5b)^3$ respectively. Therefore, the expression becomes:

$(2a)^3 + (5b)^3$

Applying the sum of cubes formula:

$(2a)^3 + (5b)^3 = (2a + 5b)\left((2a)^2 - (2a)(5b) + (5b)^2\right)$

Now, we calculate each term:

1. $(2a)^2 = 4a^2$
2. $(2a)(5b) = 10ab$
3. $(5b)^2 = 25b^2$

Thus, the expression becomes:

$8a^3 + 125b^3 = (2a + 5b)(4a^2 - 10ab + 25b^2)$

This is the fully factored form of the given expression.

Would you like more details or have any questions?

Related Questions:

1. How do you factor the difference of cubes?
2. What is the general formula for factoring a sum of cubes?
3. Can the expression $8a^3 + 125b^3$ have any real roots?
4. What are the applications of factoring in solving equations?
5. How does factoring help in simplifying algebraic expressions?
6. How do you identify a sum of cubes?
7. What is the process of factoring quadratic expressions?
8. Can the sum of cubes formula be applied to negative terms?

Tip: When factoring expressions, always check if they can be rewritten in a recognizable pattern such as a sum or difference of cubes or squares. This can simplify the problem significantly.