College

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 :

To simplify the expression [tex]\( 3(x+2)(x^2 - x - 8) \)[/tex], let's go through the steps in detail:

1. Expand the Inner Terms:
First, we distribute [tex]\( x + 2 \)[/tex] with [tex]\( x^2 - x - 8 \)[/tex]:

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

2. Use the Distributive Property ([tex]\( a(b + c) = ab + ac \)[/tex]):
We'll expand it by distributing each term in [tex]\( x + 2 \)[/tex] with [tex]\( x^2 - x - 8 \)[/tex]:

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

3. Calculate Each Distribution Independently:
- Distribute [tex]\( x \)[/tex]:

[tex]\[
x(x^2 - x - 8) = x^3 - x^2 - 8x
\][/tex]

- Distribute [tex]\( 2 \)[/tex]:

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

4. Combine Like Terms:
Adding the results from the distributions:

[tex]\[
x^3 - x^2 - 8x + 2x^2 - 2x - 16
\][/tex]

Combine the [tex]\( x^2 \)[/tex] and [tex]\( x \)[/tex] terms:

[tex]\[
x^3 + (2x^2 - x^2) + (-8x - 2x) - 16
\][/tex]

Simplifies to:

[tex]\[
x^3 + x^2 - 10x - 16
\][/tex]

So,

[tex]\[
(x + 2)(x^2 - x - 8) = x^3 + x^2 - 10x - 16
\][/tex]

5. Multiply the Result by 3:
Finally, we multiply the entire expression by 3 as given in the original problem:

[tex]\[
3(x^3 + x^2 - 10x - 16) = 3x^3 + 3x^2 - 30x - 48
\][/tex]

Thus, the simplified expression is:

[tex]\[
3(x + 2)(x^2 - x - 8) = \boxed{3x^3 + 3x^2 - 30x - 48}
\][/tex]

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