Answer
True.
Work Step by Step
To find a polynomial function for $f_{x}(x, y)$ and $f_{y}(x, y)$ we have to find the anti-derivative of this given functions first.
So, $\int f_x(x,y) dx= \int (3x^{2} + y^{2} + 2y)dx=x^3+xy^2+2xy+C(y)$ .........(1)
and $\int f_y(x,y)dy = \int (2xy + 2y)dy=xy^2+y^2+C(x)$ ............(2)
Now, from (1) and (2), the polynomial function is
$f(x, y) = x^{3} + xy^{2} + y^{2}$
So there is a polynomial checking the given conditions, therefore the statement is true.
