Mistake in The Problem / Your Post
First, there are mistakes in the decomposition. The proper equation should be:
$$
\frac{x-1}{(x-1)(x-2)^{2}}
=
\frac{A}{x-1}+
\frac{B}{x-2}+
\frac{C}{(x-2)^{2}}
$$
from which we have
$$
\frac{x-1}{(x-1)(x-2)^{2}}
=
\frac{A(x-2)^{2}+B(x-1)(x-2)+C(x-1)}{(x-1)(x-2)^{2}}
$$
Comparing The Numerators
Because the denominators on the LHS and RHS are equal, the numerators must be equal as well:
$$
x-1
=
(A+B)x^{2}+(-4A-3B+C)x+(4A+2B-C)
$$
which can be written in matrix-vector notations:
$$
\begin{Bmatrix}
x^{2} \\ x \\ 1
\end{Bmatrix}^{\top}
\begin{Bmatrix}
0 \\ 1 \\ -1
\end{Bmatrix}
=
\begin{Bmatrix}
x^{2} \\ x \\ 1
\end{Bmatrix}^{\top}
\begin{bmatrix}
1 & 1 & 0 \\
-4 & -3 & 1 \\
4 & 2 & -1
\end{bmatrix}
\begin{Bmatrix}
A \\ B \\ C
\end{Bmatrix}
$$
and the solution is
$$
\begin{Bmatrix}
A \\ B \\ C
\end{Bmatrix}
=
\begin{bmatrix}
1 & 1 & 0 \\
-4 & -3 & 1 \\
4 & 2 & -1
\end{bmatrix}^{-1}
\begin{Bmatrix}
0 \\ 1 \\ -1
\end{Bmatrix}
$$
i.e. $A=B=0$ and $C=1$