Yes, that is what I get. I did this in a slightly different way than Topsquark suggests. In particular, I did not use the MacLaurin polynomial for \(\displaystyle e^x\).
Instead I used the fact that the MacLaurin series for g is \(\displaystyle g(0)+ g'(0)x+ \frac{g''(0)}{2}x^2+ …\). To determine the coefficient of \(\displaystyle x^2\) we only need to calculate g''(0)/2.
Since \(\displaystyle g(x)= e^{f(x)}\), \(\displaystyle g'(x)= f'(x)e^{f(x)}\) and then \(\displaystyle g''(x)= f''(x)e^{f(x)}+ (f'(x))^2e^{f(x)}= (f''(x)+ (f'(x))^2)e^f(x)\).
In particular \(\displaystyle g''(0)= (f''(0)+ (f'(0))^2)e^{f(0)}\)
We are given that \(\displaystyle f(x)= 2+ 3x+ x^2+ \frac{1}{3}x^3+ \cdot\cdot\cdot\)
so f(0)= 2.
\(\displaystyle f'(x)= 3+ 2x+ x^2+ \cdot\cdot\cdot\) so f'(0)= 3.
\(\displaystyle f''(x)= 2+ 2x\cdot\cdot\cdot\) so f''(0)= 2.
The coefficient of \(\displaystyle x^2\) in the MacLaurin series for g(x) is \(\displaystyle \frac{g''(0)}{2}=\frac{(2+ 3^2)e^{2}}{2}= \frac{2+ 9}{2}e^2= \frac{11}{2}e^2\).