It looks like the vector field is
[tex]\vec f(x,y) = (y^2-6x) \, \vec\imath + 2xy \, \vec\jmath[/tex]
[tex]\vec f[/tex] is conservative if we can find a scalar function [tex]f(x,y)[/tex] whose gradient is [tex]\vec f[/tex]. This entails solving the partial differential equations
[tex]\dfrac{\partial f}{\partial x} = y^2 - 6x[/tex]
[tex]\dfrac{\partial f}{\partial y} = 2xy[/tex]
Integrate both sides of the second PDE with respect to y :
[tex]\displaystyle \int \frac{\partial f}{\partial y} \, dy = \int 2xy \, dy \implies f(x,y) = xy^2 + g(x)[/tex]
Differentiate with respect to x and solve for [tex]g(x)[/tex] :
[tex]\dfrac{\partial f}{\partial x} = y^2 - 6x = y^2 + \dfrac{dg}{dx} \implies \frac{dg}{dx} = -6x \implies g(x) = -3x^2+C[/tex]
It follows that [tex]\vec f[/tex] is indeed conservative with potential function
[tex]f(x,y) = xy^2 - 3x^2 + C[/tex]
