#### Answer

$m=\left\{ -4,-2 \right\}$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To solve the given equation, $
-4|m+3|+7=3
,$ isolate first the absolute value expression. Then use the definition of absolute value equality. Do checking of the solution/s.
$\bf{\text{Solution Details:}}$
Using the properties of equality, the given equation is equivalent to
\begin{array}{l}\require{cancel}
-4|m+3|+7=3
\\\\
-4|m+3|=3-7
\\\\
-4|m+3|=-4
\\\\
\dfrac{-4|m+3|}{-4}=\dfrac{-4}{-4}
\\\\
|m+3|=1
.\end{array}
Since for any $c\gt0$, $|x|=c$ implies $x=c \text{ or } x=-c,$ the equation above is equivalent to
\begin{array}{l}\require{cancel}
m+3=1
\\\\\text{OR}\\\\
m+3=-1
.\end{array}
Solving each equation results to
\begin{array}{l}\require{cancel}
m+3=1
\\\\
m=1-3
\\\\
m=-2
\\\\\text{OR}\\\\
m+3=-1
\\\\
m=-1-3
\\\\
m=-4
.\end{array}
If $m=-2,$ then
\begin{array}{l}\require{cancel}
-4|m+3|+7=3?
\\\\
-4|-2+3|+7=3?
\\\\
-4|1|+7=3?
\\\\
-4(1)+7=3?
\\\\
-4+7=3?
\\\\
3=3
\text{ (TRUE)}
.\end{array}
If $m=-4,$ then
\begin{array}{l}\require{cancel}
-4|m+3|+7=3?
\\\\
-4|-4+3|+7=3?
\\\\
-4|-1|+7=3?
\\\\
-4(1)+7=3?
\\\\
-4+7=3?
\\\\
3=3
\text{ (TRUE)}
.\end{array}
Hence, $
m=\left\{ -4,-2 \right\}
.$