Answer
$f(t) = 2(t-4)^2-44 $
Work Step by Step
The conversion of the function to the standard vertex form is self explanatory. All mathematical steps are shown below. $$
\begin{aligned}
f(t)&=2 t^2-16 t-12\\
& = 2\left(t^2-8 t\right) -12 \\
& = 2\left[t^2-8 t +\left( \frac{8}{2} \right)^2 \right]-12-2\cdot \left( \frac{8}{2} \right)^2 \\
& =2(t^2-8 t+4^2)^2-12-2\cdot 4^2\\
&= 2(t-4)^2-44.
\end{aligned}
$$ We got: $$ f(t) = 2(t-4)^2-44.$$
