Answer
As $\nabla \times \textbf F = 0$, so, $\textbf F$ is conservative
Work Step by Step
$\textbf F(x,y) = 15x^2y^2 \textbf i + 10x^3y \textbf j $
$\nabla \times \textbf F =\begin{vmatrix}
\textbf i & \textbf j & \textbf k\\
\frac{\delta}{\delta x} & \frac{\delta}{\delta y}& \frac{\delta}{\delta z} \\
15x^2y^2 & 10x^3y & 0\\
\end{vmatrix} = 0\textbf i + 0\textbf j + (30x^2y -30x^2y )\textbf k = 0
\\$
As $\nabla \times \textbf F = 0$, so, $\textbf F$ is conservative
