How to align these two expressions, one has one more number?

You have to remove the 1 from the \phantom, since it shows up explicitly outside the \phantom. I replace it with {} placeholder.

\documentclass{article}

\begin{document}

\[\begin{array}{c}
     (-(a-1)^2-1)x+(-(a-1)^2-1)\\
     (-(a-1)^2-1)x+\phantom{(-(a-1)^2-{})}1
\end{array}\]

\end{document}

In response to a comment from the OP, I leave this also, to show what can be done when the \phantom and residual term do not share common digits.

\documentclass{article}
\usepackage{mathtools}
\begin{document}

\[\begin{array}{c}
     (-(a-1)^2-1)x+(-(a-1)^2-1)\\
     (-(a-1)^2-1)x+\phantom{(-(a-1)^2-1)}\mathllap{5-4+2^2}
\end{array}\]

\end{document}

Add more alignment points:

\documentclass{article}
\usepackage{amsmath}
\usepackage{array}

\begin{document}

\[
\begin{array}{@{} l @{}>{{}}c<{{}}@{} r @{}}
     (-(a-1)^2-1)x &+& (-(a-1)^2-1)\\
     (-(a-1)^2-1)x &+& 1
\end{array}
\]

\end{document}