Consider a Process $X(t)$ were $t\in[0,1]$ where the process is defined to be the following... \begin{equation} X(t)=\alpha +B(t) -\beta t. \end{equation} Above I have taken the constants $\alpha$ and $\beta$ to be real-valued and assumed $B(t)$ to be a standard Brownian motion.
How would one show that this is a Gaussian process?
You have to prove that the finite-dimensional distributions are gaussians: i.e. take $0\le t_1\le t_2\le\cdots\le t_n\le1$, consider the $n$-dimensional vector $(X(t_1),\cdots,X(t_n))$ and prove that it is a gaussian vector. But this is true, since it is an affine transformation of the vector $(B(t_1),\cdots,B(t_n))$ which is gaussian, since $B$ is Brownian Motion.