Answer
$F$ is Not conservative.
Work Step by Step
When $F(x,y)=pi+qj$ is a conservative field, then throughout the $D$, then we have
$\dfrac{\partial p}{\partial y}=\dfrac{\partial q}{\partial x}$
Here, $p$ and $q$ represents the first-order partial derivatives on a domain $D$.
Let us consider $p=(xy+y^2)$ and $q=(x^2+2xy)$
The first-order partial derivatives are: $p_x=x+2y$ and $q_x=2x+2y$
Here, we can see that $\dfrac{\partial A}{\partial y} \neq \dfrac{\partial Q}{\partial x}$
Hence, $F$ is not conservative.
