Select the correct answer.

Function [tex]\( h \)[/tex] is the product of functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex].

[tex]\[
\begin{array}{l}
f(x)=2x+5 \\
g(x)=6x-9
\end{array}
\][/tex]

Which equation defines function [tex]\( h \)[/tex]?

A. [tex]\( h(x)=12x^2-4x-45 \)[/tex]

B. [tex]\( h(x)=12x-45 \)[/tex]

C. [tex]\( h(x)=12x^2-45 \)[/tex]

D. [tex]\( h(x)=12x^2+12x-45 \)[/tex]



Answer :

Let's determine the function [tex]\( h(x) \)[/tex] by finding the product of the functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex].
Given:
[tex]\[ f(x) = 2x + 5 \][/tex]
[tex]\[ g(x) = 6x - 9 \][/tex]

The product function [tex]\( h(x) \)[/tex] is:
[tex]\[ h(x) = f(x) \cdot g(x) \][/tex]
[tex]\[ h(x) = (2x + 5) \cdot (6x - 9) \][/tex]

Next, we'll expand this product:
[tex]\[ h(x) = (2x + 5)(6x - 9) \][/tex]

To expand, we use the distributive property (also known as the FOIL method for binomials):
1. Multiply the first terms:
[tex]\[ 2x \cdot 6x = 12x^2 \][/tex]
2. Multiply the outer terms:
[tex]\[ 2x \cdot (-9) = -18x \][/tex]
3. Multiply the inner terms:
[tex]\[ 5 \cdot 6x = 30x \][/tex]
4. Multiply the last terms:
[tex]\[ 5 \cdot (-9) = -45 \][/tex]

Now, combine all these products:
[tex]\[ h(x) = 12x^2 - 18x + 30x - 45 \][/tex]

Combine like terms:
[tex]\[ h(x) = 12x^2 + ( -18x + 30x ) - 45 \][/tex]
[tex]\[ h(x) = 12x^2 + 12x - 45 \][/tex]

Thus, the simplified form of [tex]\( h(x) \)[/tex] is:
[tex]\[ h(x) = 12x^2 + 12x - 45 \][/tex]

Therefore, the correct equation that defines function [tex]\( h \)[/tex] is:
[tex]\[ \boxed{D. \ h(x)=12 x^2+12 x-45} \][/tex]