To determine which expression is equivalent to the given polynomial expression [tex]\((9mn - 19m^4n) - (8m^2 + 12m^4n + 9mn)\)[/tex], let's go through the problem step-by-step.
1. Distribute the Negative Sign: Start by distributing the negative sign across the second polynomial inside the parentheses:
[tex]\[
(9mn - 19m^4n) - (8m^2 + 12m^4n + 9mn) \rightarrow (9mn - 19m^4n) - 8m^2 - 12m^4n - 9mn
\][/tex]
2. Combine Like Terms: Next, we combine like terms from the resulting expression:
Let's group the [tex]\(mn\)[/tex] terms together, the [tex]\(m^4n\)[/tex] terms together, and the [tex]\(m^2\)[/tex] term separately:
[tex]\[
9mn - 9mn - 19m^4n - 12m^4n - 8m^2
\][/tex]
3. Simplify: Simplify the combined terms:
- The [tex]\(mn\)[/tex] terms: [tex]\(9mn - 9mn = 0\)[/tex]
- The [tex]\(m^4n\)[/tex] terms: [tex]\(-19m^4n - 12m^4n = -31m^4n\)[/tex]
- The [tex]\(m^2\)[/tex] term: [tex]\(-8m^2\)[/tex] (since there is no like term to combine with)
Putting it all together, we get:
[tex]\[
0 - 31m^4n - 8m^2 = -31m^4n - 8m^2
\][/tex]
4. Conclusion: The equivalent expression is:
[tex]\[
-31m^4n - 8m^2
\][/tex]
Hence, the correct answer is A. [tex]\(-31m^4n - 8m^2\)[/tex].
