Answer
Not conservative.
Work Step by Step
Given: $F(x,y)=(xy+y^2)i+(x^2+2xy)j$
if $F(x,y)=Ai+Bj$ is a conservative field, then throughout the domain $D$, we get
$\dfrac{\partial A}{\partial y}=\dfrac{\partial B}{\partial x}$
Here, $A$ and $B$ are first-order partial derivatives on the domain $D$.
Consider $A=(xy+y^2)$ and $B=(x^2+2xy)$
Then, we have $A_x=x+2y; B=2x+2y$
Here, $\dfrac{\partial A}{\partial y} \neq \dfrac{\partial Q}{\partial x}$
Thus, $F(x,y)$ is not conservative.
