Answer
$g(x)=\frac{3}{8}\left(x+\frac{1}{6}\right)^2-\frac{85}{96} $
Work Step by Step
The conversion of the function to the standard vertex form is self explanatory. All mathematical steps are shown below. I have perform a trick by multiply both sides of the equation by 8 so that we can work with a simpler expression.
$$
\begin{aligned}
g(x) & =\frac{3}{8} x^2+\frac{1}{8} x-\frac{7}{8} \\
8 g(x) & =3 x^2+x-7 \\
& =3\left(x^2+\frac{1}{3} x\right)-7 \\
& =3\left[x^2+\frac{1}{3} x+\left(\frac{1}{6}\right)^2\right]-7-3\left(\frac{1}{6}\right)^2 \\
& =3\left(x+\frac{1}{6}\right)^2-7-\frac{3}{36} \\
& =3\left(x+\frac{1}{6}\right)^2-\frac{7 \cdot 12}{12}-\frac{1}{12} \\
& =3\left(x+\frac{1}{6}\right)^2-\frac{85}{12}.
\end{aligned}
$$ Divide the above equation by 8 to get an expression for $g(x)$.
$$
\Rightarrow \quad g(x)=\frac{3}{8}\left(x+\frac{1}{6}\right)^2-\frac{85}{96}.
$$
