Answer
It has symmetry with respect to $y$-axis
Work Step by Step
There are couple of ways to test for symmetry. The easiest and fastest way is to sketch the graph and spot the symmetry visually.
We can use graphing calculator to graph the following equation: $y=x^2+|x|$. See the image above.
As one can clearly see, it has symmetry with respect to $y$-axis.
To check it algebraically, we have to replace $x$ by $-x$ and obtain the original equation
$y=(-x)^2+|-x|$ => $y=x^2+|x|$
