Let's begin by examining the given functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
[tex]\[ f(x) = \sqrt{x^2 + 24x + 144} \][/tex]
First, we will simplify the expression inside the square root:
[tex]\[ x^2 + 24x + 144 \][/tex]
Notice that this is a perfect square trinomial:
[tex]\[ x^2 + 24x + 144 = (x + 12)^2 \][/tex]
Therefore,
[tex]\[ f(x) = \sqrt{(x + 12)^2} \][/tex]
Since the square root of a square is the absolute value,
[tex]\[ f(x) = |x + 12| \][/tex]
For simplicity in this context, we'll assume [tex]\( x \geq -12 \)[/tex]. Hence,
[tex]\[ f(x) = x + 12 \][/tex]
Now, consider the function [tex]\( g(x) \)[/tex]:
[tex]\[ g(x) = x^3 - 216 \][/tex]
To find [tex]\( f(x) + g(x) \)[/tex], we simply add [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
[tex]\[ f(x) + g(x) = (x + 12) + (x^3 - 216) \][/tex]
Combine the terms to get:
[tex]\[ f(x) + g(x) = x^3 + x + 12 - 216 \][/tex]
[tex]\[ f(x) + g(x) = x^3 + x - 204 \][/tex]
Hence, the expression equal to [tex]\( f(x) + g(x) \)[/tex] is:
[tex]\[ \boxed{x^3 + x - 204} \][/tex]
