Answer
$\color{blue}{(-\infty, -4.5) \cup (-4.5, 1) \cup (1, +\infty)}$
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:
For $f(x)$, $-4.5$ will make the denominator zero so its domain is: $(-\infty, -4.5) \cup (-4.5, +\infty)$
$g(x)$ is defined for all real numbers except $1$ (as it makes the denominator zero) so its domain is: $(-\infty,1) \cup (1, +\infty)$.
Note that:
$[(-\infty, -4.5) \cup(-4.5, +\infty)] \cap [(-\infty, 1) \cup (1, +\infty)]
\\= (-\infty, -4.5) \cup (-4.5, 1) \cup (1, +\infty)$.
Thus, the domain of the sum, difference, and product of $f(x)$ and $g(x)$ is:
$\color{blue}{(-\infty, -4.5) \cup (-4.5, 1) \cup (1, +\infty)}$