To simplify the expression \(12mn + 30m^2n^3\), we should look for common factors in the terms and factor them out.

### Steps to Simplify:

1. **Identify Common Factors:**
   - The first term is \(12mn\).
   - The second term is \(30m^2n^3\).

2. **Find the Greatest Common Factor (GCF):**
   - For the coefficients: The GCF of 12 and 30 is 6.
   - For the variables:
     - The GCF of \(m\) in \(m\) and \(m^2\) is \(m\).
     - The GCF of \(n\) in \(n\) and \(n^3\) is \(n\).

3. **Factor Out the GCF:**
   - The GCF of the entire expression is \(6mn\).

4. **Rewrite Each Term:**
   - \(12mn = 6mn \times 2\)
   - \(30m^2n^3 = 6mn \times 5mn^2\)

5. **Factor Out the GCF:**
   \[
   12mn + 30m^2n^3 = 6mn(2 + 5mn^2)
   \]

Thus, the simplified form of the expression is:

\[
6mn(2 + 5mn^2)
\]

Would you like more details or have any questions?

Here are some related questions for further exploration:

1. How do you find the greatest common factor of a polynomial?
2. What are some methods for factoring polynomials?
3. How can you check if your factorization is correct?
4. What is the difference between a monomial and a polynomial?
5. Can you simplify expressions with more than two terms?
6. How do you simplify expressions with negative exponents?
7. What is the role of variables in algebraic expressions?
8. How can you apply the distributive property in reverse to factor expressions?

**Tip:** When simplifying expressions, always look for the greatest common factor first, as it helps in factoring efficiently.

Learn how to simplify the expression 12mn + 30m²n³ with a detailed step-by-step solution. This problem covers key algebraic concepts including polynomials and factoring. Suitable for students in Grades 7-9.

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