Answer
$ k(m) = (m+2.5)^2-18.25$
Work Step by Step
The conversion of the function to the standard vertex form $k(m)=a(m-h)^2+k$ is self explanatory. All mathematical steps are shown below. $$
\begin{aligned}
k(m)&=m^2+5m-12\\
& = m^2+5 m+\left(\frac{5}{2}\right)^2 -12-\left(\frac{5}{2}\right)^2 \\
& =\left(m^2+5 m +2.5^2\right)-12-2.5^2 \\
& =(m+2.5)^2-18.25.
\end{aligned}
$$ The vertex form of the given function is $$k(m)=(m+2.5)^2-18.25.$$
