[tex]\mathbf f[/tex] is conservative if we can find a scalar function [tex]f[/tex] such that [tex]\nabla f=\mathbf f[/tex]. This is equivalent to solving the system of PDEs,
[tex]\dfrac{\partial f}{\partial x}=2x-6y[/tex]
[tex]\dfrac{\partial f}{\partial y}=-6x+10y-9[/tex]
Integrate both sides of the first PDF with respect to [tex]x[/tex]:
[tex]f(x,y)=x^2-6xy+g(y)[/tex]
where [tex]g[/tex] is some function independent of [tex]x[/tex]. Then differentiatng with respect to [tex]y[/tex], we have
[tex]\dfrac{\partial f}{\partial y}=-6x+\dfrac{\mathrm dg}{\mathrm dy}=-6x+10y-9[/tex]
[tex]\implies\dfrac{\mathrm dg}{\mathrm dy}=10y-9\implies g(y)=5y^2-9y+C[/tex]
and so [tex]\mathbf f[/tex] is indeed conservative, with
[tex]f(x,y)=x^2-6xy+5y^2-9y+C[/tex]
