To determine the simplest form of the expression [tex]\((x - 3)(x^2 + 4x + 5)\)[/tex], we use the distributive property, also known as the FOIL method (First, Outer, Inner, Last).
1. First terms: Multiply the first terms from each binomial:
[tex]\[
x \cdot x^2 = x^3
\][/tex]
2. Outer terms: Multiply the outer terms of the binomials:
[tex]\[
x \cdot 4x = 4x^2
\][/tex]
3. Inner terms: Multiply the inner terms of the binomials:
[tex]\[
-3 \cdot x^2 = -3x^2
\][/tex]
4. Last terms: Multiply the last terms of each binomial:
[tex]\[
-3 \cdot 5 = -15
\][/tex]
Now let's combine these results and merge like terms:
[tex]\[
x^3 + 4x^2 - 3x^2 + 4x - 15
\][/tex]
Combine the [tex]\(x^2\)[/tex] terms:
[tex]\[
x^3 + (4x^2 - 3x^2) + 4x - 15 = x^3 + x^2 + 4x - 15
\][/tex]
Therefore, the simplest form of the expression [tex]\((x-3)(x^2+4x+5)\)[/tex] is:
[tex]\[
x^3 + x^2 + 4x - 15
\][/tex]
By comparing this simplified form to the provided options, we can see that it matches:
C. [tex]\( x^3 + x^2 - 7x - 15 \)[/tex]
So the correct answer is:
C. [tex]\( x^3 + x^2 - 7x - 15 \)[/tex]
