Answer
$m= 9, m= 3$
Work Step by Step
We start by completing the square in line $1$ of the equation by dividing the coefficient of the $12m$ term by $2$ and squaring it and adding the result on both sides.
$$
\begin{aligned}
m^2-12m & = -27 \\
m^2-12m+\left(\frac{12}{2}\right)^2 & =-27+\left(\frac{12}{2}\right)^2 \\
m^2-12m+6^2 & =-27+6^2 \\
(m-6)^2 & =9.
\end{aligned}
$$ Take the square root: $$
\begin{array}{rl}
(m-6)^2 & =9 \\
m-6 & = \pm \sqrt{9} \\
m-6 & = 3\\
m-6= 3 &\ \textbf{or}\ \ \ m-6=-3\\
m=6+3= 9 & \ \textbf{or}\ \ \ m= 6-3= 3.
\end{array}
$$ Check $$\begin{aligned}
9^2-12(9) & \stackrel{?}{=} -27 \\
-27& = -27\checkmark
\end{aligned}$$ $$\begin{aligned}
3^2-12(3) & \stackrel{?}{=} -27 \\
-27& = -27\checkmark.
\end{aligned}$$ The solution is: $$m= 9, m= 3.$$
