What is the result of multiplying these polynomial numbers?

(3m²n + mn - 12) . (2m²n - mn² + 7)

To multiply the polynomials \((3m^2n + mn - 12)\) and \((2m^2n - mn^2 + 7)\), we will use the distributive property (also known as FOIL when multiplying binomials, but for polynomials, we expand all terms).

The expression we need to expand is:

\[
(3m^2n + mn - 12)(2m^2n - mn^2 + 7)
\]

Now, distribute each term in the first polynomial by every term in the second polynomial:

1. **Multiply \(3m^2n\) by each term in the second polynomial:**

\[
3m^2n \cdot 2m^2n = 6m^4n^2
\]
\[
3m^2n \cdot (-mn^2) = -3m^3n^3
\]
\[
3m^2n \cdot 7 = 21m^2n
\]

2. **Multiply \(mn\) by each term in the second polynomial:**

\[
mn \cdot 2m^2n = 2m^3n^2
\]
\[
mn \cdot (-mn^2) = -m^2n^3
\]
\[
mn \cdot 7 = 7mn
\]

3. **Multiply \(-12\) by each term in the second polynomial:**

\[
-12 \cdot 2m^2n = -24m^2n
\]
\[
-12 \cdot (-mn^2) = 12mn^2
\]
\[
-12 \cdot 7 = -84
\]

Now, let's combine like terms:

\[
6m^4n^2 - 3m^3n^3 + 21m^2n + 2m^3n^2 - m^2n^3 + 7mn - 24m^2n + 12mn^2 - 84
\]

Grouping like terms:

- \(6m^4n^2\) (no other \(m^4n^2\) terms)
- \((-3m^3n^3 + 2m^3n^2)\)
- \((21m^2n - 24m^2n - m^2n^3)\)
- \((7mn + 12mn^2)\)
- \(-84\)

Simplifying:

\[
6m^4n^2 + (-3m^3n^3 + 2m^3n^2) + (-3m^2n - m^2n^3) + (7mn + 12mn^2) - 84
\]

Final result:

\[
6m^4n^2 - 3m^3n^3 + 2m^3n^2 - 3m^2n - m^2n^3 + 7mn + 12mn^2 - 84
\]

Would you like to go over any specific step in more detail?

### Related Questions:
1. How do you factor a polynomial after multiplication?
2. How can you find the degree of a polynomial?
3. What is the difference between monomials, binomials, and polynomials?
4. How do you use the FOIL method for binomials?
5. What are common mistakes in polynomial multiplication?

**Tip:** Always combine like terms carefully to simplify the result when multiplying polynomials.

What is the result of multiplying these polynomial numbers? (3m²n + mn - 12) . (2m²n - mn² + 7)

Discover how to multiply polynomials using the distributive property with this step-by-step solution to (3m²n + mn - 12)(2m²n - mn² + 7). Understand key algebraic concepts and techniques. Ideal for high school students (Grades 10-12).

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