ARMA one step ahead forecast and forecast error

#1
For a ARMA(1,1) process with constant \(\theta\) is
\(X_t=\alpha X_{t-1}+\theta +Z_t+\beta Z_{t-1}\) where \( Z_T\) is white noise with mean 0 and variance\(\sigma ^2\).
1)Find the one step ahead forecast
2)Find the expected value and variance of one step ahead forecast error

Here's what I did:
1)\(\hat X_{t+1}=\hat \alpha X_t+\theta+\hat \beta Z_t\).
I would like to know if this is the correct one step ahead forecast.
Also does \(\theta\) becomes \(\hat \theta\)?

I think not because it is a constant.
For second part:
error=\(X_{t+1} -\)\( \hat X_{t+1}=X_t(\alpha-\hat \alpha)+Z_t(\beta-\hat \beta)+Z_{t+1}\).

Is this forecast error correct?

Then E(error)=0
and V(error)=\((\alpha-\hat \alpha)^2 V(X_t)+(\beta-\hat \beta)^2\sigma^2+\sigma^2\)

Can someone please help me to understand this