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Which expression is equivalent to the given polynomial expression?
[tex]\ \textless \ br/\ \textgreater \ \left(-4a^2 - 3b\right) + \left(-2ab - a^2 + b^2\right) + \left(-b^2 + 6ab\right)\ \textless \ br/\ \textgreater \ [/tex]

A. [tex]-3a^2 + 4ab + 3b[/tex]
B. [tex]-5a^2 + 2b^2 + 8ab + 3b[/tex]
C. [tex]-3a^2 + 2b^2 + 8ab + 3b[/tex]
D. [tex]-5a^2 + 4ab - 3b[/tex]



Answer :

GinnyAnswer
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]

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