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

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

In this case:

$125a^3 = (5a)^3 \quad \text{and} \quad 8b^3 = (2b)^3.$

So, the expression becomes:

$125a^3 + 8b^3 = (5a + 2b)\left((5a)^2 - (5a)(2b) + (2b)^2\right).$

Now, simplify the factors:

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

This is the fully factorized form of the expression.

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Here are 5 related questions to expand your understanding:

1. How do you factorize $27x^3 + 64y^3$ using the sum of cubes formula?
2. What is the difference between the sum and difference of cubes formulas?
3. How would you factorize $64a^3 + 1$?
4. How do you determine when to use the sum of cubes versus difference of cubes?
5. What are some real-life applications of the sum of cubes factorization?

Tip: Memorizing the sum and difference of cubes formulas helps you quickly factor similar expressions in algebraic problems.