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Answer :
We start with the expression
[tex]$$
3(x+2)(x^2 - x - 8).
$$[/tex]
Step 1. Expand the product inside the parentheses.
First, expand
[tex]$$
(x+2)(x^2 - x - 8).
$$[/tex]
We use the distributive property (also known as the FOIL method for binomials) to multiply each term in [tex]$(x+2)$[/tex] by each term in [tex]$(x^2-x-8)$[/tex]:
[tex]\[
\begin{aligned}
(x+2)(x^2 - x - 8) &= x\cdot (x^2) + x\cdot (-x) + x\cdot (-8) \\
&\quad + 2\cdot (x^2) + 2\cdot (-x) + 2\cdot (-8) \\
&= x^3 - x^2 - 8x + 2x^2 - 2x - 16.
\end{aligned}
\][/tex]
Now, combine like terms:
- Combine [tex]$-x^2$[/tex] and [tex]$2x^2$[/tex]:
[tex]$$
-x^2 + 2x^2 = x^2.
$$[/tex]
- Combine [tex]$-8x$[/tex] and [tex]$-2x$[/tex]:
[tex]$$
-8x - 2x = -10x.
$$[/tex]
So the expanded result is
[tex]$$
x^3 + x^2 - 10x - 16.
$$[/tex]
Step 2. Multiply the result by 3.
Now, multiply the entire expression by 3:
[tex]$$
3(x^3 + x^2 - 10x - 16).
$$[/tex]
Distributing the 3:
[tex]\[
\begin{aligned}
3(x^3 + x^2 - 10x - 16) &= 3x^3 + 3x^2 - 30x - 48.
\end{aligned}
\][/tex]
Thus, the simplified expression is
[tex]$$
\boxed{3x^3 + 3x^2 - 30x - 48}.
$$[/tex]
[tex]$$
3(x+2)(x^2 - x - 8).
$$[/tex]
Step 1. Expand the product inside the parentheses.
First, expand
[tex]$$
(x+2)(x^2 - x - 8).
$$[/tex]
We use the distributive property (also known as the FOIL method for binomials) to multiply each term in [tex]$(x+2)$[/tex] by each term in [tex]$(x^2-x-8)$[/tex]:
[tex]\[
\begin{aligned}
(x+2)(x^2 - x - 8) &= x\cdot (x^2) + x\cdot (-x) + x\cdot (-8) \\
&\quad + 2\cdot (x^2) + 2\cdot (-x) + 2\cdot (-8) \\
&= x^3 - x^2 - 8x + 2x^2 - 2x - 16.
\end{aligned}
\][/tex]
Now, combine like terms:
- Combine [tex]$-x^2$[/tex] and [tex]$2x^2$[/tex]:
[tex]$$
-x^2 + 2x^2 = x^2.
$$[/tex]
- Combine [tex]$-8x$[/tex] and [tex]$-2x$[/tex]:
[tex]$$
-8x - 2x = -10x.
$$[/tex]
So the expanded result is
[tex]$$
x^3 + x^2 - 10x - 16.
$$[/tex]
Step 2. Multiply the result by 3.
Now, multiply the entire expression by 3:
[tex]$$
3(x^3 + x^2 - 10x - 16).
$$[/tex]
Distributing the 3:
[tex]\[
\begin{aligned}
3(x^3 + x^2 - 10x - 16) &= 3x^3 + 3x^2 - 30x - 48.
\end{aligned}
\][/tex]
Thus, the simplified expression is
[tex]$$
\boxed{3x^3 + 3x^2 - 30x - 48}.
$$[/tex]
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