Answer:
[tex]\sf B)\ f(x) \cdot g(x) = x^4 + 6x^3 - 12x - 76[/tex]
Step-by-step explanation:
Given functions:
[tex]\sf f(x) = \sqrt{x^2+12x+36}\\[/tex]
[tex]\sf g(x) = x^3 - 12[/tex]
[tex]\sf f(x) \cdot g(x) = \left(\sqrt{x^2+12x+36}\right)\times (x^3 - 12) \\\\\ \textsf{[rewrite the radicand using the perfect square trinomial rule: $a^2+2ab+b=(a+b)^2$]}\\\\[/tex]
[tex]\sf f(x) \times g(x) = \left(\sqrt{(x+6)^2}\right) \times \left(x^3 - 12\right)\textsf{[take the square root of the perfect square trinomial]}\\\\\sf f(x) \cdot g(x) = \left(x+6}\right) \times \left(x^3 - 12\right) \textsf{[multiply]}\\\\f(x) \cdot g(x) = x\left(x^3 - 12\right) + 6 \left(x^3 - 12\right) \textsf{[simplify]}\\\\f(x) \cdot g(x) = x^4 - 12x + 6x^3 - 76 \textsf{ [rewrite in descending order]}\\\\f(x) \cdot g(x) = x^4 + 6x^3 - 12x - 76[/tex]
