#### Answer

$\text{all real numbers except }$ $
m=\left\{ -4,-3 \right\}$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
The domain of the given rational function, $
h(m)=\dfrac{4m^2+2m-9}{m^2+7m+12}
,$ are the values of $
m
$ which will NOT make the denominator equal to $0.$
$\bf{\text{Solution Details:}}$
The values of $
m
$ that will make the denominator equal to $0$ are
\begin{array}{l}\require{cancel}
m^2+7m+12=0
.\end{array}
Using the FOIL Method, which is given by $(a+b)(c+d)=ac+ad+bc+bd,$ the equation above is equivalent to
\begin{array}{l}\require{cancel}
(m+3)(m+4)=0
.\end{array}
Equating each factor to zero (Zero Product Property), then
\begin{array}{l}\require{cancel}
m+3=0
\\\\\text{OR}\\\\
m+4=0
.\end{array}
Solving each equation results to
\begin{array}{l}\require{cancel}
m+3=0
\\\\
m=-3
\\\\\text{OR}\\\\
m+4=0
\\\\
m=-4
.\end{array}
Hence, the domain is the set of $
\text{all real numbers except }$ $
m=\left\{ -4,-3 \right\}
.$