Answer
Neither even nor odd.
Work Step by Step
$f$ is even if the following equation holds for all $x$ and $-x$ in the domain of $f$, $f(x)=f(-x)$.
$f$ is odd if the following equation holds for all $x$ and $-x$ in the domain of $f$, $-f(x)=f(-x)$.
Let's evaluate $f(-x)$:
$f(-x)=\frac{(-x)^2}{2(-x)+1}=\frac{x^2}{-2x+1}$
Note that:
$f(-x) \ne f(x)$ therefore the function is not even.
$f(-x) \ne -f(x)$ therefore the function is not odd.
