Answer
$ g(t) = (t-4)^2-36 $
Work Step by Step
The conversion of the function to the standard vertex form $g(t)=a(t-h)^2+k$ s self explanatory. All mathematical steps are shown below.$$
\begin{aligned}
g(t) & = t^2-8 t-20\\
& = t^2-8 t+\left(\frac{8}{2}\right)^2 -20-\left(\frac{8}{2}\right)^2 \\
& =\left(t^2-8 t+4^2\right)-20-4^2 \\
& =(t-4)^2-36.
\end{aligned}
$$ The vertex form of the given function is $$ g(t) = (t-4)^2-36.$$
