To simplify the expression [tex]\((x - 3)(x^2 + 4x + 5)\)[/tex], follow these steps:
1. Distribute each term in the first binomial to each term in the second trinomial:
[tex]\[
(x - 3)(x^2 + 4x + 5) = x \cdot (x^2 + 4x + 5) - 3 \cdot (x^2 + 4x + 5)
\][/tex]
2. Distribute [tex]\( x \)[/tex] across [tex]\( (x^2 + 4x + 5) \)[/tex]:
[tex]\[
x \cdot (x^2 + 4x + 5) = x \cdot x^2 + x \cdot 4x + x \cdot 5 = x^3 + 4x^2 + 5x
\][/tex]
3. Distribute [tex]\(-3\)[/tex] across [tex]\( (x^2 + 4x + 5) \)[/tex]:
[tex]\[
-3 \cdot (x^2 + 4x + 5) = -3 \cdot x^2 - 3 \cdot 4x - 3 \cdot 5 = -3x^2 - 12x - 15
\][/tex]
4. Combine the results from both distributions:
[tex]\[
(x - 3)(x^2 + 4x + 5) = x^3 + 4x^2 + 5x - 3x^2 - 12x - 15
\][/tex]
5. Combine like terms:
[tex]\[
x^3 + (4x^2 - 3x^2) + (5x - 12x) - 15 = x^3 + x^2 - 7x - 15
\][/tex]
So, the simplest form of the expression is:
[tex]\[
x^3 + x^2 - 7x - 15
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{\text{C. } x^3 + x^2 - 7x - 15}
\][/tex]
