Answer :
Let's simplify the expression given in the question: [tex]\(-4x^3 - 12x^3 + 9x^2\)[/tex].
### Step 1: Combine Like Terms
- The like terms in the expression are [tex]\(-4x^3\)[/tex] and [tex]\(-12x^3\)[/tex]. We can combine these by adding their coefficients:
[tex]\[
-4x^3 - 12x^3 = (-4 - 12)x^3 = -16x^3
\][/tex]
- The term [tex]\(9x^2\)[/tex] does not have a like term to combine with, so it remains the same.
### Step 2: Write the Simplified Expression
- Combining the results from step 1, we have:
[tex]\[
-16x^3 + 9x^2
\][/tex]
Now, let's compare this to the given options to find the equivalent expression:
1. [tex]\(x^8\)[/tex]
2. [tex]\(-7x^8\)[/tex]
3. [tex]\(-8x^3 + 9x^2\)[/tex]
4. [tex]\(-16x^3 + 9x^2\)[/tex]
5. [tex]\(-16x^6 + 9x^2\)[/tex]
The option that matches the simplified expression [tex]\(-16x^3 + 9x^2\)[/tex] is:
Option 4: [tex]\(-16x^3 + 9x^2\)[/tex]
Therefore, the answer is [tex]\(-16x^3 + 9x^2\)[/tex].
### Step 1: Combine Like Terms
- The like terms in the expression are [tex]\(-4x^3\)[/tex] and [tex]\(-12x^3\)[/tex]. We can combine these by adding their coefficients:
[tex]\[
-4x^3 - 12x^3 = (-4 - 12)x^3 = -16x^3
\][/tex]
- The term [tex]\(9x^2\)[/tex] does not have a like term to combine with, so it remains the same.
### Step 2: Write the Simplified Expression
- Combining the results from step 1, we have:
[tex]\[
-16x^3 + 9x^2
\][/tex]
Now, let's compare this to the given options to find the equivalent expression:
1. [tex]\(x^8\)[/tex]
2. [tex]\(-7x^8\)[/tex]
3. [tex]\(-8x^3 + 9x^2\)[/tex]
4. [tex]\(-16x^3 + 9x^2\)[/tex]
5. [tex]\(-16x^6 + 9x^2\)[/tex]
The option that matches the simplified expression [tex]\(-16x^3 + 9x^2\)[/tex] is:
Option 4: [tex]\(-16x^3 + 9x^2\)[/tex]
Therefore, the answer is [tex]\(-16x^3 + 9x^2\)[/tex].
