To simplify the expression [tex]\((x-3)(x^2 + 4x + 5)\)[/tex], let's go through the steps methodically.
1. Distribute [tex]\(x-3\)[/tex] across [tex]\(x^2 + 4x + 5\)[/tex]:
[tex]\[
(x-3)(x^2 + 4x + 5)
\][/tex]
2. Expand using the distributive property (also known as the FOIL method for binomials):
[tex]\[
= x \cdot (x^2 + 4x + 5) - 3 \cdot (x^2 + 4x + 5)
\][/tex]
3. Distribute [tex]\(x\)[/tex] and [tex]\(-3\)[/tex] across the terms inside the parentheses:
[tex]\[
= x \cdot x^2 + x \cdot 4x + x \cdot 5 - 3 \cdot x^2 - 3 \cdot 4x - 3 \cdot 5
\][/tex]
4. Simplify each term:
[tex]\[
= x^3 + 4x^2 + 5x - 3x^2 - 12x - 15
\][/tex]
5. Combine like terms:
[tex]\[
= x^3 + (4x^2 - 3x^2) + (5x - 12x) - 15
\][/tex]
[tex]\[
= x^3 + x^2 - 7x - 15
\][/tex]
So the simplified form of the expression [tex]\((x-3)(x^2 + 4x + 5)\)[/tex] is:
[tex]\[
x^3 + x^2 - 7x - 15
\][/tex]
Checking the answer options:
- A. [tex]\(x^3 + 7x^2 + 7x + 15\)[/tex]
- B. [tex]\(x^3 + x^2 + 17x + 15\)[/tex]
- C. [tex]\(x^3 + x^2 - 7x - 15\)[/tex]
- D. [tex]\(x^3 - 7x^2 - 17x - 15\)[/tex]
Clearly, the simplified expression matches option C.
Therefore, the correct answer is:
C. [tex]\(x^3 + x^2 - 7x - 15\)[/tex]
