we are given $n$ independent standard Gaussian random variables $x_i \sim N(0,1)$ and deterministic integer coefficients $c_i\geq 1$. What is the moment generating function of $Z=(\sum_{i=1}^n c_ix_i)^2$.
So since $x_i$ are normally distributed variables if I multiply them by a constant I obtain the normally distributed variables $x_i^\prime \sim N(0, c_i^2)$. The sum of normally distributed random variables is distributed as well, and therefore $\sum_{i=1}^n c_ix_i \sim N(0, \sum c_i^2)$. This means that $Z$ is a chi-squared random variable with one degree of freedom. If $Z$ is centered, then the moment generating function is given by $M(t)=(1-2t)^{-1/2}$. Is $Z$ centered?
A linear combination of independent normal variables is normal. The square of a normal is Gamma. Hence, your variable has a Gamma distribution (see a Gamma random variable is a sum of squared normal random variables). On the same page you can find the mgf of the Gamma (it exists only for certain values of the parameter).