The expression \( x^4 + 125x \) can be factored or simplified depending on what you're looking for. Since it's not a standard form for complete factoring, we can factor out the common term \( x \):

\[
x^4 + 125x = x(x^3 + 125)
\]

Now, notice that \( x^3 + 125 \) is a sum of cubes, which can be factored using the sum of cubes formula:

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

Here, \( a = x \) and \( b = 5 \) because \( 125 = 5^3 \). Applying the formula:

\[
x^3 + 125 = (x + 5)(x^2 - 5x + 25)
\]

Thus, the full factorization of \( x^4 + 125x \) is:

\[
x^4 + 125x = x(x + 5)(x^2 - 5x + 25)
\]

Would you like further details or explanations?

Here are 5 related questions to extend your understanding:
1. How do you factor other sums and differences of cubes?
2. What is the geometric interpretation of factoring polynomials?
3. How does the sum of cubes differ from the difference of cubes formula?
4. Can you apply this factorization method to higher-degree polynomials?
5. How do imaginary numbers come into play when factoring sums of cubes?

**Tip:** Always check for common factors first when simplifying algebraic expressions!

Learn how to factor the polynomial x^4 + 125x by extracting common terms and applying the sum of cubes formula. This step-by-step explanation breaks down the factoring process for advanced algebra students in Grades 10-12.

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