Moment-generating function of $Z:=X_1X_2+X_3X_4$

Question

Let $X_1,X_2,X_3,X_4$ be four indipendent random variable with normal distribution of mean 0 and variance 1. The exercise asks me to calculate the moment-generating function of $X_1X_2$. I was able to do it and I found that $$M_{X_1X_2}(z) = \displaystyle\frac{1}{\sqrt{1-z^2}}$$ I believe that this result is correct since I found a similar question here. The exercise goes on and ask to prove that the moment-generating function of $Z:=X_1X_2+X_3X_4$ is $$M_Z(t)= \displaystyle\frac{1}{1+t^2}$$ What I write was: $$M_Z(t)=\Bbb E[e^{tZ}]=\Bbb E[e^{tX_1X_2+tX_3X_4}]=\Bbb E[e^{tX_1X_2}e^{tX_3X_4}] \overset{indip.}{=} \Bbb E[e^{tX_1X_2}]\Bbb E[e^{tX_3X_4}]=\displaystyle \left(\frac{1}{\sqrt{1-t^2}}\right)^2=\frac{1}{1-t^2}$$ Am I missing something or there is an error in the exercise's text?