Thank you so much for the help Alan!! I didn't think of doing that before but I understand why that is helpful now.

If my work is correct, the sum of the coefficients of the terms in g(x) that have even degree is 21,556,128.5

\(g(1)=f(1)^{16}=(-\frac{3}{2}+\frac{1}{2}+4)^{16}=3^{16}=43,046,721 \)

\(g(-1)=f(-1)^{16}=(-\frac{3}{2}-\frac{1}{2}+4)^{16}=2^{16}=65,536\)

\(\frac{g(1)+g(-1)}{2}=\frac{43,046,721+65,536}{2}=\frac{43,112,257}{2}=21,556,128.5\)

Thanks again for helping me solve this!