Let's simplify the given polynomial expression step-by-step:
[tex]\[
\left(-4 a^2 - 3 b\right) + \left(-2 a b - a^2 + b^2\right) + \left(-b^2 + 6 a b\right)
\][/tex]
1. Group the like terms together:
- For [tex]\(a^2\)[/tex]:
[tex]\[
-4 a^2 - a^2 = -5 a^2
\][/tex]
- For [tex]\(b^2\)[/tex]:
[tex]\[
b^2 - b^2 = 0
\][/tex]
- For [tex]\(ab\)[/tex]:
[tex]\[
-2 a b + 6 a b = 4 a b
\][/tex]
- For [tex]\(b\)[/tex]:
[tex]\[
-3 b
\][/tex]
2. Combine the like terms to form the simplified polynomial:
[tex]\[
-5 a^2 + 4 a b - 3 b
\][/tex]
Now, let's match this result with the given answer choices:
A. [tex]\(-3 a^2 + 4 a b + 3 b\)[/tex]
B. [tex]\(-5 a^2 + 2 b^2 + 8 a b + 3 b\)[/tex]
C. [tex]\(-3 a^2 + 2 b^2 + 8 a b + 3 b\)[/tex]
D. [tex]\(-5 a^2 + 4 a b - 3 b\)[/tex]
The simplified polynomial [tex]\(-5 a^2 + 4 a b - 3 b\)[/tex] matches exactly with option D.
Therefore, the correct answer is:
[tex]\[
\boxed{D}
\][/tex]
