To determine the correct equation defining the function [tex]\( h(x) \)[/tex], which is the product of functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex], follow these steps:
1. Recall the functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
[tex]\[
f(x) = 4x - 6
\][/tex]
[tex]\[
g(x) = 5x + 9
\][/tex]
2. Form the product [tex]\( h(x) \)[/tex] by multiplying [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
[tex]\[
h(x) = f(x) \cdot g(x)
\][/tex]
[tex]\[
h(x) = (4x - 6)(5x + 9)
\][/tex]
3. Expand the product using the distributive property (FOIL method):
[tex]\[
h(x) = (4x - 6)(5x + 9)
\][/tex]
[tex]\[
= 4x \cdot 5x + 4x \cdot 9 - 6 \cdot 5x - 6 \cdot 9
\][/tex]
4. Calculate each term individually:
[tex]\[
4x \cdot 5x = 20x^2
\][/tex]
[tex]\[
4x \cdot 9 = 36x
\][/tex]
[tex]\[
-6 \cdot 5x = -30x
\][/tex]
[tex]\[
-6 \cdot 9 = -54
\][/tex]
5. Combine all the terms:
[tex]\[
h(x) = 20x^2 + 36x - 30x - 54
\][/tex]
6. Simplify the expression by combining like terms:
[tex]\[
36x - 30x = 6x
\][/tex]
[tex]\[
h(x) = 20x^2 + 6x - 54
\][/tex]
Thus, the equation that defines [tex]\( h(x) \)[/tex] is:
[tex]\[
h(x) = 20x^2 + 6x - 54
\][/tex]
Among the given choices, the correct answer is:
D. [tex]\( h(x) = 20x^2 + 6x - 54 \)[/tex]
