factoring poles / zeros: off by constant gain compared with textbook
Spectral factorization is not unique, unless you specify further constraints.

So let your given $H(z)H(z^-1)$ be (taking from your derivation) :

$$
\begin{align}
H(z)H(z^{-1})&amp;= \frac{ \frac{5}{4} - \frac{1}{2}z - \frac{1}{2}z^{-1}  }{ \frac{10}{9}- \frac{1}{3}z - \frac{1}{3}z^-1 } \tag{1}\\ 
&amp;= \frac{\left(1-\frac{1}{2}z^{-1}\right)(1-2z^{-1})}{\left(1-\frac{1}{3}z^{-1}\right)(1-3z^{-1})}
\end{align}
$$

Then all of the following are valid possibilities that satisf Eq.(1)

$$
H_1(z) =\frac{1-\frac{1}{2}z^{-1}}{1-\frac{1}{3}z^{-1}}
$$

$$
H_2(z) =\frac{1-2z^{-1}}{1-3z^{-1}}
$$

$$
H_3(z) =\frac{1-\frac{1}{2}z^{-1}}{1-3z^{-1}}
$$

$$
H_4(z) =\frac{1-2z^{-1}}{1-\frac{1}{3}z^{-1}}
$$

So you must have further constraints that make one of them as a unique answer. Such as minimum-phase, maximum phase, causal, stable, bothe etc...