spectral factorization method
The question is not so clear. I am not sure what other types of examples you consider. In this particular case, if you didn't get the hint notice that:

$$\begin{align}
c(1+\alpha D)(1+\alpha D^{-1}) &amp;=  (c+\alpha^2c)+\alpha cD^{-1}+\alpha cD \\&amp;=0.371+0.09D^{-1} +.09D
\end{align}$$

which gives you
$$\left\{\begin{matrix}
c(1+\alpha^2) = 0.371\\
\alpha c=0.09
\end{matrix}\right.
\Rightarrow \frac{1+\alpha^2}{\alpha}=\frac{0.37}{0.09}$$
and you will actually have two solutions

$$\left\{\begin{matrix}
\alpha=3.8634, c=    0.0233\\
\alpha=0.2588, c=  0.3477
\end{matrix}\right.
$$

See if the second answer is also valid...