Answer
$$
\left(\frac{x+2}{6}\right)^{2}+\left(\frac{y+12}{3}\right)^{2}=1.
$$
Work Step by Step
The ellipse
$$
\left(\frac{x-8}{6}\right)^{2}+\left(\frac{y+4}{3}\right)^{2}=1
$$ has the center $(8,-4)$. Translating the center to $(-2,-12)$, leads to the new ellipse
$$
\left(\frac{x+2}{6}\right)^{2}+\left(\frac{y+12}{3}\right)^{2}=1.
$$
