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.