Sure, let's simplify the expression [tex]\((x + 4)\left(x^2 - 6x + 3\right)\)[/tex] step-by-step.
First, let's distribute [tex]\((x + 4)\)[/tex] across each term in the polynomial [tex]\((x^2 - 6x + 3)\)[/tex]:
[tex]\[
(x + 4)\left(x^2 - 6x + 3\right) = (x + 4) \cdot x^2 + (x + 4) \cdot (-6x) + (x + 4) \cdot 3
\][/tex]
Now, we distribute each term separately.
1. Distribute [tex]\(x^2\)[/tex]:
[tex]\[
(x + 4) \cdot x^2 = x \cdot x^2 + 4 \cdot x^2 = x^3 + 4x^2
\][/tex]
2. Distribute [tex]\(-6x\)[/tex]:
[tex]\[
(x + 4) \cdot (-6x) = x \cdot (-6x) + 4 \cdot (-6x) = -6x^2 - 24x
\][/tex]
3. Distribute [tex]\(3\)[/tex]:
[tex]\[
(x + 4) \cdot 3 = x \cdot 3 + 4 \cdot 3 = 3x + 12
\][/tex]
Now, combine all these results together:
[tex]\[
x^3 + 4x^2 - 6x^2 - 24x + 3x + 12
\][/tex]
Next, combine like terms:
- For [tex]\(x^2\)[/tex] terms: [tex]\(4x^2 - 6x^2 = -2x^2\)[/tex]
- For [tex]\(x\)[/tex] terms: [tex]\(-24x + 3x = -21x\)[/tex]
- The constant term is [tex]\(12\)[/tex]
So, the simplified expression is:
[tex]\[
x^3 - 2x^2 - 21x + 12
\][/tex]
Thus, the correct answer is:
[tex]\[
x^3 - 2x^2 - 21x + 12
\][/tex]
