To determine which expression is the simplest form of [tex]\( -\left(2x^3 + x^2\right) + 3\left(x^3 - 4x^2\right) \)[/tex], let's go through the steps to simplify the given expression:
1. Expand the given expression:
The expression to simplify is:
[tex]\[
-\left(2x^3 + x^2\right) + 3\left(x^3 - 4x^2\right)
\][/tex]
2. Distribute the negative sign and the scalar:
Distribute the negative sign inside the first term and the scalar [tex]\( 3 \)[/tex] inside the second term:
[tex]\[
-(2x^3 + x^2) = -2x^3 - x^2
\][/tex]
[tex]\[
3(x^3 - 4x^2) = 3x^3 - 12x^2
\][/tex]
3. Combine the expanded terms:
Now add the two results together:
[tex]\[
-2x^3 - x^2 + 3x^3 - 12x^2
\][/tex]
4. Combine like terms:
Combine the [tex]\( x^3 \)[/tex] terms:
[tex]\[
-2x^3 + 3x^3 = x^3
\][/tex]
Combine the [tex]\( x^2 \)[/tex] terms:
[tex]\[
-x^2 - 12x^2 = -13x^2
\][/tex]
5. Write the simplified expression:
The simplest form of the given expression is:
[tex]\[
x^3 - 13x^2
\][/tex]
6. Match the simplified expression with the given options:
- Option A: [tex]\( x^3 - 13x^2 \)[/tex]
- Option B: [tex]\( x^3 - 3x^2 \)[/tex]
- Option C: [tex]\( 5x^3 - 12x^2 \)[/tex]
- Option D: [tex]\( 5x^3 - 11x^2 \)[/tex]
The simplified expression [tex]\( x^3 - 13x^2 \)[/tex] matches Option A.
Therefore, the correct answer is:
[tex]\[
\boxed{\text{A}}
\][/tex]
