Answer
$ f(x) = (x+3)^2-1 $
Work Step by Step
The conversion of the function to the standard vertex form is self explanatory. All mathematical steps are shown below.$$
\begin{aligned}
f(x) & = x^2+6 x+8 \\
& = x^2+6 x+\left(\frac{6}{2}\right)^2 +8-\left(\frac{6}{2}\right)^2 \\
& =\left(x^2+6 x+3^2\right)+8-3^2 \\
& =(x+3)^2-1.
\end{aligned}
$$
