Answer :
To simplify the expression [tex]\(3(x+2)(x^2-x-8)\)[/tex], we'll follow a step-by-step process.
1. Distribute the 3 to the first part of the expression:
Start by distributing the 3 in front to the expression [tex]\( (x+2) \)[/tex], but as the question is asked to simplify the whole expression it'll be equivalent if expanded later.
2. Use the distributive property:
To simplify [tex]\((x+2)(x^2-x-8)\)[/tex], apply the distributive property:
[tex]\[
x(x^2-x-8) + 2(x^2-x-8)
\][/tex]
3. Multiply each term:
- For [tex]\(x(x^2-x-8)\)[/tex]:
[tex]\[
x \cdot x^2 = x^3
\][/tex]
[tex]\[
x \cdot (-x) = -x^2
\][/tex]
[tex]\[
x \cdot (-8) = -8x
\][/tex]
- For [tex]\(2(x^2-x-8)\)[/tex]:
[tex]\[
2 \cdot x^2 = 2x^2
\][/tex]
[tex]\[
2 \cdot (-x) = -2x
\][/tex]
[tex]\[
2 \cdot (-8) = -16
\][/tex]
4. Combine like terms:
Combine all terms from the above steps in one expression:
[tex]\[
x^3 - x^2 - 8x + 2x^2 - 2x - 16
\][/tex]
[tex]\[
= x^3 + (-x^2 + 2x^2) + (-8x - 2x) - 16
\][/tex]
[tex]\[
= x^3 + x^2 - 10x - 16
\][/tex]
5. Combine this result with the outer 3:
Finally, multiply the entire expression by 3:
[tex]\[
3(x^3 + x^2 - 10x - 16)
\][/tex]
[tex]\[
= 3x^3 + 3x^2 - 30x - 48
\][/tex]
Therefore, the simplified expression is:
[tex]\[ 3x^3 + 3x^2 - 30x - 48 \][/tex]
This matches with one of the provided answer choices.
1. Distribute the 3 to the first part of the expression:
Start by distributing the 3 in front to the expression [tex]\( (x+2) \)[/tex], but as the question is asked to simplify the whole expression it'll be equivalent if expanded later.
2. Use the distributive property:
To simplify [tex]\((x+2)(x^2-x-8)\)[/tex], apply the distributive property:
[tex]\[
x(x^2-x-8) + 2(x^2-x-8)
\][/tex]
3. Multiply each term:
- For [tex]\(x(x^2-x-8)\)[/tex]:
[tex]\[
x \cdot x^2 = x^3
\][/tex]
[tex]\[
x \cdot (-x) = -x^2
\][/tex]
[tex]\[
x \cdot (-8) = -8x
\][/tex]
- For [tex]\(2(x^2-x-8)\)[/tex]:
[tex]\[
2 \cdot x^2 = 2x^2
\][/tex]
[tex]\[
2 \cdot (-x) = -2x
\][/tex]
[tex]\[
2 \cdot (-8) = -16
\][/tex]
4. Combine like terms:
Combine all terms from the above steps in one expression:
[tex]\[
x^3 - x^2 - 8x + 2x^2 - 2x - 16
\][/tex]
[tex]\[
= x^3 + (-x^2 + 2x^2) + (-8x - 2x) - 16
\][/tex]
[tex]\[
= x^3 + x^2 - 10x - 16
\][/tex]
5. Combine this result with the outer 3:
Finally, multiply the entire expression by 3:
[tex]\[
3(x^3 + x^2 - 10x - 16)
\][/tex]
[tex]\[
= 3x^3 + 3x^2 - 30x - 48
\][/tex]
Therefore, the simplified expression is:
[tex]\[ 3x^3 + 3x^2 - 30x - 48 \][/tex]
This matches with one of the provided answer choices.
