Here's another possibility, but it requires mathJax v2.0 (because of the [t] option for \begin{array}).
t-1\; \mathop{\overline{\smash{\Big)}
\begin{array}[t]{l}
\,t^3+t^2-1\\
\dfrac{t^3+t^2}{\phantom{t^3}-t^2}\\
\hphantom{t^3} \dfrac{{}-t^2-t}{\phantom{{}-t^2-}t}\\
\hphantom{t^3-t^2-{}\!} \dfrac{t+1}{\phantom{t}-1}
\end{array}
}}^{\hphantom{\Big)}\,\textstyle t^2-t\phantom{^2}-1\phantom{{}+0}}
It doesn't have as much space hacking, so it works a bit more consistently across output renderers and fonts. It also is more closely related to your original. But again it requires MathJax v2.0 (e.g., the v2.0-candidate branch from my fork of MathJax).