Answer :
To simplify the expression [tex]\(3(x+2)(x^2-x-8)\)[/tex], we need to expand it step by step. Let's go through the process:
1. Distribute the first term:
[tex]\[
3(x + 2)(x^2 - x - 8) = 3 \cdot \left[(x + 2)(x^2 - x - 8)\right]
\][/tex]
2. Expand [tex]\( (x + 2)(x^2 - x - 8) \)[/tex]:
We can expand this using the distributive property:
[tex]\[
(x + 2)(x^2 - x - 8) = x(x^2 - x - 8) + 2(x^2 - x - 8)
\][/tex]
3. Distribute each term separately:
- For [tex]\( x(x^2 - x - 8) \)[/tex], distribute [tex]\( x \)[/tex]:
[tex]\[
x^3 - x^2 - 8x
\][/tex]
- For [tex]\( 2(x^2 - x - 8) \)[/tex], distribute [tex]\( 2 \)[/tex]:
[tex]\[
2x^2 - 2x - 16
\][/tex]
4. Add all the terms together:
Combine the results from the expansions:
[tex]\[
x^3 - x^2 - 8x + 2x^2 - 2x - 16
\][/tex]
Simplify by combining like terms:
[tex]\[
x^3 + (-x^2 + 2x^2) + (-8x - 2x) - 16
\][/tex]
[tex]\[
x^3 + x^2 - 10x - 16
\][/tex]
5. Multiply by 3 (as per the original expression):
Now distribute the 3:
[tex]\[
3(x^3 + x^2 - 10x - 16)
\][/tex]
- This gives:
[tex]\[
3x^3 + 3x^2 - 30x - 48
\][/tex]
The simplified expression matches the second option:
[tex]\[
3x^3 + x^2 - 10x - 16
\][/tex]
Therefore, the answer is [tex]\(\boxed{3x^3 + x^2 - 10x - 16}\)[/tex].
1. Distribute the first term:
[tex]\[
3(x + 2)(x^2 - x - 8) = 3 \cdot \left[(x + 2)(x^2 - x - 8)\right]
\][/tex]
2. Expand [tex]\( (x + 2)(x^2 - x - 8) \)[/tex]:
We can expand this using the distributive property:
[tex]\[
(x + 2)(x^2 - x - 8) = x(x^2 - x - 8) + 2(x^2 - x - 8)
\][/tex]
3. Distribute each term separately:
- For [tex]\( x(x^2 - x - 8) \)[/tex], distribute [tex]\( x \)[/tex]:
[tex]\[
x^3 - x^2 - 8x
\][/tex]
- For [tex]\( 2(x^2 - x - 8) \)[/tex], distribute [tex]\( 2 \)[/tex]:
[tex]\[
2x^2 - 2x - 16
\][/tex]
4. Add all the terms together:
Combine the results from the expansions:
[tex]\[
x^3 - x^2 - 8x + 2x^2 - 2x - 16
\][/tex]
Simplify by combining like terms:
[tex]\[
x^3 + (-x^2 + 2x^2) + (-8x - 2x) - 16
\][/tex]
[tex]\[
x^3 + x^2 - 10x - 16
\][/tex]
5. Multiply by 3 (as per the original expression):
Now distribute the 3:
[tex]\[
3(x^3 + x^2 - 10x - 16)
\][/tex]
- This gives:
[tex]\[
3x^3 + 3x^2 - 30x - 48
\][/tex]
The simplified expression matches the second option:
[tex]\[
3x^3 + x^2 - 10x - 16
\][/tex]
Therefore, the answer is [tex]\(\boxed{3x^3 + x^2 - 10x - 16}\)[/tex].
