Answer
Odd, origin.
Work Step by Step
We know that if a function is odd, then $f(-x)=-f(x).$
We know that if a function is eben, then $f(-x)=f(x).$
Hence we plug in $-x$ to see what happens.
$g(-x)=\frac{2(-x)^3}{(-x)^4+1}=\frac{-2x^3}{x^4+1}=-g(x)$, hence the function is odd.
We know that if a graph is symmetric to the origin, then the points $(a,b)$ and $(-a,-b)$ will be on the graph, hence $f(-b)=-f(b)$, hence this is the same as the function being odd.
