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The expression provided is \( u^3 + 125v^3 \), which is a sum of cubes. 

The formula for factoring a sum of cubes is:

\[
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
\]

In this case:
- \( a = u \)
- \( b = 5v \), because \( 125v^3 = (5v)^3 \).

Using the sum of cubes formula, we get:

\[
u^3 + 125v^3 = (u + 5v)(u^2 - 5uv + 25v^2)
\]

So, the completely factored form of the expression is:

\[
(u + 5v)(u^2 - 5uv + 25v^2)
\]

Would you like any additional details or explanations on this factorization? 

Here are some related questions to expand your understanding:
1. How do you factor the difference of cubes?
2. Can you apply the sum of cubes formula to other numbers, like \(8x^3 + 27y^3\)?
3. What is the difference between factoring a sum of cubes and a perfect square trinomial?
4. How would you apply this formula if there were coefficients attached to both terms, like \( 2u^3 + 250v^3 \)?
5. How do you verify that your factorization is correct?

**Tip**: When factoring sums or differences of cubes, always identify the cube roots of both terms first to simplify the process!

Completely factor the expression u^3 + 125v^3.

Learn how to completely factor the expression u^3 + 125v^3 using the sum of cubes formula. This problem covers essential algebraic concepts and techniques for factoring cubic equations, suitable for high school students in Grades 9-12.

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