First, let's start by carefully breaking down the given expression:
[tex]\[
(9mn - 19m^4n) - (8m^2 + 12m^4n + 9mn)
\][/tex]
We need to distribute the negative sign across the second set of parentheses:
[tex]\[
= 9mn - 19m^4n - (8m^2 + 12m^4n + 9mn)
\][/tex]
[tex]\[
= 9mn - 19m^4n - 8m^2 - 12m^4n - 9mn
\][/tex]
Notice that [tex]\(9mn\)[/tex] and [tex]\(-9mn\)[/tex] cancel each other out:
[tex]\[
= -19m^4n - 8m^2 - 12m^4n
\][/tex]
Combine like terms:
[tex]\[
= (-19m^4n - 12m^4n) - 8m^2
\][/tex]
[tex]\[
= -31m^4n - 8m^2
\][/tex]
Thus, the simplified expression is:
[tex]\[
m^2(-31m^2n - 8)
\][/tex]
So, the correct answer is:
C. [tex]\(-31m^4n - 8m^2\)[/tex]
