4. Given that $f(x), g(x)$ are functions defined on $\mathbf{R}$, the graph of $f(x)$ is symmetric about the line $x=1$, the graph of $g(x)$ is symmetric about the point $(1,-2)$, and $f(x)+g(x)=9^{x}+x^{3}+1$, then the value of $f(2) g(2)$ is $\qquad$ .
According to the problem, $\left\{\begin{array}{l}f(x)=f(2-x), \\ g(x)+g(2-x)=-4\end{array} \Rightarrow\left\{\begin{array}{l}f(2)=f(0), \\ g(2)+g(0)=-4 .\end{array}\right.\right.$
And $\left\{\begin{array}{l}f(2)+g(2)=9^{2}+2^{3}+1=90, \\ f(0)+g(0)=9^{0}+1=2\end{array} \Rightarrow\left\{\begin{array}{l}f(2)+g(2)=90, \\ f(2)-g(2)=6\end{array} \Rightarrow\left\{\begin{array}{l}f(2)=48, \\ g(2)=42 .\end{array}\right.\right.\right.$
Therefore, $f(2) g(2)=48 \times 42=(45+3)(45-3)=2025-9=2016$.