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]
