To find [tex]\( f(x) \cdot g(x) \)[/tex] where [tex]\( f(x) = x^2 + 3x - 4 \)[/tex] and [tex]\( g(x) = x + 5 \)[/tex]:
1. Write the expressions for [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
[tex]\[
f(x) = x^2 + 3x - 4
\][/tex]
[tex]\[
g(x) = x + 5
\][/tex]
2. Setup the product [tex]\( f(x) \cdot g(x) \)[/tex]:
[tex]\[
f(x) \cdot g(x) = (x^2 + 3x - 4) \cdot (x + 5)
\][/tex]
3. Distribute each term in [tex]\( f(x) \)[/tex] to every term in [tex]\( g(x) \)[/tex]:
- [tex]\( (x^2 + 3x - 4) \cdot (x + 5) \)[/tex]
[tex]\[
= x^2 \cdot (x + 5) + 3x \cdot (x + 5) - 4 \cdot (x + 5)
\][/tex]
4. Multiply each term:
- [tex]\( x^2 \cdot x = x^3 \)[/tex]
- [tex]\( x^2 \cdot 5 = 5x^2 \)[/tex]
- [tex]\( 3x \cdot x = 3x^2 \)[/tex]
- [tex]\( 3x \cdot 5 = 15x \)[/tex]
- [tex]\( -4 \cdot x = -4x \)[/tex]
- [tex]\( -4 \cdot 5 = -20 \)[/tex]
Combine the results:
[tex]\[
x^3 + 5x^2 + 3x^2 + 15x - 4x - 20
\][/tex]
5. Combine like terms:
- [tex]\( 5x^2 + 3x^2 = 8x^2 \)[/tex]
- [tex]\( 15x - 4x = 11x \)[/tex]
Simplify:
[tex]\[
x^3 + 8x^2 + 11x - 20
\][/tex]
Thus, the product [tex]\( f(x) \cdot g(x) \)[/tex] is:
[tex]\[
x^3 + 8x^2 + 11x - 20
\][/tex]
The correct option is:
[tex]\[
\boxed{x^3 + 8x^2 + 11x - 20}
\][/tex]
