Algebra

Completing the square in $x$ and $y$, we get

$\begin{align*} -4 (x + 1)^2 + (y - 1)^2 = 6. \end{align*}$

We can re-write this as

$\begin{align*} \frac{(y - 1)^2}{6} - \frac{(x + 1)^2}{3/2} = 1. \end{align*}$

The center of the hyperbola is at $\boxed{(-1,1)}$