Answer
$\color{blue}{(-\infty, -2) \cup (-2, +\infty)}$
Work Step by Step
RECALL:
The domain of the sum, difference, and product of $f(x)$ and $g(x)$ is the common elements of the domains of the two functions.
The domain of the respective functions are:
$f(x)$ is defined for all real numbers so its domain is: $(-\infty, +\infty)$
$g(x)$ is defined for all real numbers except $-2$ so its domain is: $(-\infty,-2) \cup (-2 +\infty)$.
Note that:
$[(-\infty, -2) \cup(-2, +\infty)] \cap (-\infty, +\infty)
\\= (-\infty, -2) \cup (2, +\infty)$.
Thus, the domain of the sum, difference, and product of $f(x)$ and $g(x)$ is:
$\color{blue}{(-\infty, -2) \cup (-2, +\infty)}$