#### Answer

$m=\left\{ -2-3i,-2+3i \right\}$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To find the non real complex solutions of the given equation, $
m^2+4m+13=0
,$ use the properties of equality to express the given equation in the form $x^2+bx=c.$ Then complete the square by adding $\left(\dfrac{b}{2} \right)^2$ to both sides. Factor the left side then take the square root (Square Root Property) of both sides. Then use the properties of radicals and use $i=\sqrt{-1}.$ Finally, simplify the radical and isolate the variable.
$\bf{\text{Solution Details:}}$
Using the properties of equality, in the form $x^2+bx=c,$ the given equation is equivalent to
\begin{array}{l}\require{cancel}
m^2+4m=-13
.\end{array}
In the equation above, $b=
4
.$ Substituting $b$ in the expression $\left( \dfrac{b}{2} \right)^2,$ then
\begin{array}{l}\require{cancel}
\left( \dfrac{4}{2} \right)^2
\\\\=
\left( 2 \right)^2
\\\\=
4
.\end{array}
Adding $\left(\dfrac{b}{2} \right)^2$ to both sides of the equation above to complete the square, the equation becomes
\begin{array}{l}\require{cancel}
m^2+4m+4=-13+4
\\\\
(m+2)^2=-9
.\end{array}
Taking the square root of both sides (Square Root Property), the equation above is equivalent to
\begin{array}{l}\require{cancel}
m+2=\pm\sqrt{-9}
.\end{array}
Using the Product Rule of radicals which is given by $\sqrt[m]{x}\cdot\sqrt[m]{y}=\sqrt[m]{xy},$ the equation above is equivalent to\begin{array}{l}\require{cancel}
m+2=\pm\sqrt{-1}\cdot\sqrt{9}
.\end{array}
Using $i=\sqrt{-1},$ the equation above is equivalent to
\begin{array}{l}\require{cancel}
m+2=\pm i\sqrt{9}
.\end{array}
Writing the radicand as an expression that contains a factor that is a perfect power of the given index and then extracting the root of that factor, the equation above is equivalent to
\begin{array}{l}\require{cancel}
m+2=\pm i\sqrt{(3)^2}
\\\\
m+2=\pm i(3)
\\\\
m+2=\pm 3i
.\end{array}
Using the properties of equality to isolate the variable, the equation above is equivalent to
\begin{array}{l}\require{cancel}
m=-2\pm 3i
.\end{array}
Hence, $
m=\left\{ -2-3i,-2+3i \right\}
.$