High School

Simplify the expression [tex]$3(x+2)\left(x^2-x-8\right)$[/tex]:

A. [tex]$3x^3 + 3x^2 - 30x - 48$[/tex]
B. [tex]$3x^3 + x^2 - 10x - 16$[/tex]
C. [tex]$3x^3 - 30x^2 - 12x - 48$[/tex]
D. [tex]$3x^3 - 4x^2 - 30x - 48$[/tex]

Answer :

Sure, let's simplify the expression step-by-step:

Given expression:
[tex]\[ 3(x + 2)(x^2 - x - 8) \][/tex]

Let's break it down into smaller steps:

1. Expand the inner expression first:

[tex]\[
(x + 2)(x^2 - x - 8)
\][/tex]

2. Distribute each term in [tex]\((x + 2)\)[/tex] across [tex]\((x^2 - x - 8)\)[/tex]:

[tex]\[
(x + 2) \times (x^2 - x - 8)
\][/tex]

This will give us:
[tex]\[
x \times (x^2 - x - 8) + 2 \times (x^2 - x - 8)
\][/tex]

3. Simplify the products:
- For [tex]\(x \times (x^2 - x - 8)\)[/tex]:
[tex]\[
x(x^2 - x - 8) = x^3 - x^2 - 8x
\][/tex]

- For [tex]\(2 \times (x^2 - x - 8)\)[/tex]:
[tex]\[
2(x^2 - x - 8) = 2x^2 - 2x - 16
\][/tex]

4. Combine the results:
[tex]\[
x^3 - x^2 - 8x + 2x^2 - 2x - 16
\][/tex]

5. Combine like terms:
[tex]\[
x^3 + (2x^2 - x^2) + (-8x - 2x) - 16
\][/tex]
[tex]\[
x^3 + x^2 - 10x - 16
\][/tex]

6. Multiply by the coefficient outside the parentheses:
[tex]\[
3(x^3 + x^2 - 10x - 16)
\][/tex]

7. Distribute the 3:
[tex]\[
3x^3 + 3x^2 - 30x - 48
\][/tex]

So, the simplified expression is:
[tex]\[ 3x^3 + 3x^2 - 30x - 48 \][/tex]

Therefore, the correct answer is:
[tex]\[
\boxed{3x^3 + 3x^2 - 30x - 48}
\][/tex]

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