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The problem asks you to divide a polynomial by a monomial:

\[
\frac{-12u^2x^4 + 15u^5x^3 + 21u^7x^4}{3u^5x^3}
\]

### Steps to simplify:

1. **Separate each term**: Divide each term in the numerator by \(3u^5x^3\).
   - First term: \(\frac{-12u^2x^4}{3u^5x^3}\)
   - Second term: \(\frac{15u^5x^3}{3u^5x^3}\)
   - Third term: \(\frac{21u^7x^4}{3u^5x^3}\)

2. **Simplify each fraction**:
   - For the first term: \(\frac{-12u^2x^4}{3u^5x^3} = -4u^{2-5}x^{4-3} = -4u^{-3}x^1 = -4x/u^3\)
   - For the second term: \(\frac{15u^5x^3}{3u^5x^3} = 5\)
   - For the third term: \(\frac{21u^7x^4}{3u^5x^3} = 7u^{7-5}x^{4-3} = 7u^2x^1 = 7u^2x\)

3. **Combine all simplified terms**:
   - Final result: \[
   -\frac{4x}{u^3} + 5 + 7u^2x
   \]

Let me know if you need any clarifications!

Here are five related questions to practice:
1. How do you divide a polynomial by a binomial?
2. What happens when a term in the numerator has no corresponding variables in the denominator?
3. How do you handle negative exponents when simplifying?
4. What is the distributive property in relation to polynomial division?
5. Can you simplify the expression further if additional terms are included?

**Tip:** Always subtract exponents when dividing like variables, and watch for negative exponents!

Divide the polynomial: (-12u^2x^4 + 15u^5x^3 + 21u^7x^4) by (3u^5x^3) and simplify.

This detailed guide explains how to simplify the polynomial expression (-12u^2x^4 + 15u^5x^3 + 21u^7x^4) divided by 3u^5x^3. Learn key algebraic concepts such as dividing polynomials, simplifying exponents, and the laws of exponents. Suitable for students in Grades 10-12.

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