Answer
$ h(x) = 3(x+2)^2+12$
Work Step by Step
The conversion of the function to the standard vertex form $h(x)=a(x-h)^2+k$ is self explanatory. All mathematical are show below. $$
\begin{aligned}
h(x) & = 3 x^2+12x +24 \\
& =3\left(x^2+\frac{12}{3} x\right)+24 \\
& =3\left[x^2+4 x+\left(\frac{4}{2}\right)^2\right] +24-3\left(\frac{4}{2}\right)^2 \\
& =3\left(x^2+4 x+2^2\right)+24-3\cdot 2^2 \\
& =3(x+2)^2+12.
\end{aligned}
$$ The vertex form of the given function is $$h(x)=3(x+2)^2+12.$$
