$X_t$ = $3/2X_{t-1} - 1/2X_{t-2} + 1/2e_t - e_{t-1}$
Write the model into an ARMA form, determine if it is stationary if it is invertible and determine p and q. If it can be reduced, write the function of the ARMA model in the simplest form.
Actually, I do not know how to deal with $1/2e_t$, because the general form of ARMA model, $e_t$ do not have a coefficient.
You can simply multiply the whole equation by 2. This gives you \begin{equation} 2 X_t = 3X_{t-1} - X_{t-2} + e_t - 2e_{t-1} \end{equation}
But I guess you might need the coefficient of $X_t$ t be 1. In this case you can redefine the error process.
Assume that $e_t$ is white noise with zero mean and variance $\lambda$. Then you can define a new process $\varepsilon_t = 0.5 e_t$. The new process is white noise with zero mean and variance $0.25\lambda$.
You end up with the following equivalent model \begin{equation} X_t = \frac{3}{2}X_{t-1} - \frac{1}{2}X_{t-2} + \varepsilon_t - \frac{1}{2} \varepsilon _{t-1} \end{equation}