Let’s solve the problem step-by-step.
Given the functions:
[tex]\[ f(x) = 7x + 4 \][/tex]
[tex]\[ g(x) = 9x + 6 \][/tex]
We need to find the product of the functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex], denoted as [tex]\( fg(x) \)[/tex].
To find [tex]\( fg(x) \)[/tex], we multiply the two given functions:
[tex]\[ fg(x) = f(x) \cdot g(x) \][/tex]
[tex]\[ fg(x) = (7x + 4) \cdot (9x + 6) \][/tex]
Next, we use the distributive property to expand the expression:
[tex]\[ fg(x) = 7x \cdot 9x + 7x \cdot 6 + 4 \cdot 9x + 4 \cdot 6 \][/tex]
Now, let's calculate each term separately:
[tex]\[ 7x \cdot 9x = 63x^2 \][/tex]
[tex]\[ 7x \cdot 6 = 42x \][/tex]
[tex]\[ 4 \cdot 9x = 36x \][/tex]
[tex]\[ 4 \cdot 6 = 24 \][/tex]
Combining all these terms together, we get:
[tex]\[ fg(x) = 63x^2 + 42x + 36x + 24 \][/tex]
Next, we combine the like terms:
[tex]\[ fg(x) = 63x^2 + (42x + 36x) + 24 \][/tex]
[tex]\[ fg(x) = 63x^2 + 78x + 24 \][/tex]
So, the expression for [tex]\( fg(x) \)[/tex] is:
[tex]\[ 63x^2 + 78x + 24 \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{63x^2 + 78x + 24} \][/tex]
So, the answer choice is:
[tex]\[ \mathbf{D.} \: 63x^2 + 78x + 24 \][/tex]
