To determine the expression for [tex]\( f(x) \cdot g(x) \)[/tex], we begin by analyzing the given functions:
1. [tex]\( f(x) = \sqrt{x^2 + 12x + 36} \)[/tex]
2. [tex]\( g(x) = x^3 - 12 \)[/tex]
First, simplify [tex]\( f(x) \)[/tex]:
[tex]\[ f(x) = \sqrt{x^2 + 12x + 36} \][/tex]
Notice that [tex]\( x^2 + 12x + 36 \)[/tex] is a perfect square trinomial:
[tex]\[ x^2 + 12x + 36 = (x + 6)^2 \][/tex]
Therefore,
[tex]\[ f(x) = \sqrt{(x + 6)^2} \][/tex]
Since [tex]\( \sqrt{(x + 6)^2} = |x + 6| \)[/tex], and assuming for the domain of [tex]\( x \geq -6 \)[/tex], [tex]\( |x + 6| = x + 6 \)[/tex]. Thus,
[tex]\[ f(x) = x + 6 \][/tex]
Next, calculate the product [tex]\( f(x) \cdot g(x) \)[/tex]:
[tex]\[ f(x) \cdot g(x) = (x + 6) \cdot (x^3 - 12) \][/tex]
Now, distribute [tex]\( x + 6 \)[/tex] across [tex]\( x^3 - 12 \)[/tex]:
[tex]\[ (x + 6)(x^3 - 12) = x \cdot x^3 + x \cdot (-12) + 6 \cdot x^3 + 6 \cdot (-12) \][/tex]
[tex]\[ = x^4 - 12x + 6x^3 - 72 \][/tex]
Therefore, the expression for [tex]\( f(x) \cdot g(x) \)[/tex] is:
[tex]\[ f(x) \cdot g(x) = x^4 + 6x^3 - 12x - 72 \][/tex]
Thus, the correct answer is:
C. [tex]\( x^4 + 6x^3 - 12x - 72 \)[/tex]
