I am not sure how to proceed with the question below. I know the theorem that states what the mgf is... I'm just not sure how to properly apply it. Essentially, from the question below, I know that the matrix will be of rank 3, it will have diagonal entries of 4, 4, 1, and 0... However, I am unsure how to formally write this as a moment generating function. Help would be great. Thanks.

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1) if $Z \sim N(0,1)$, the MGF of $Z^2$ is $\frac{1}{\sqrt{1-2t}}$.
2) if $X_1, X_2, \ldots, X_k$ are independent random variables with MGF's $M_1(t), M_2(t), \ldots, M_k(t)$ and $a_1, a_2, \ldots, a_k$ are scalars, the MGF of $a_1 X_1 + a_2 X_2 + \ldots + a_k X_k$ is $M_1(a_1 t) M_2(a_2 t) \ldots M_k(a_k t)$