Answer
The domain of $H(x)$ is the set of all real numbers $x$ except $2$ and $-4$.
In set notation: $\text{Domain:} \{x|x \neq 2, \hspace{5pt} x \neq -4\}$
Work Step by Step
Rational functions are of the form
$$R(x)=\dfrac{p(x)}{q(x)}$$
The domain of the rational function is the set of all real numbers except those for which the denominator $q(x)$ is $0$.
$\text{Set the denominator equal to zero}$ then solve:
$$(x-2)(x+4)= 0\\[3mm] \text{Using the zero-product property:} \\[5mm] \\x-2=0 \text{ or } x+4=0
\\x =2 \hspace{10pt} \text{or} \hspace{10pt} x=-4$$
The domain of $H(x)$ is the set of all real numbers $x$ except $2$ and $-4$.
$\text{Domain:} \{x|x \neq 2, \hspace{5pt} x \neq -4\}$
