Answer
$$\dfrac{-1}{2}$$
Work Step by Step
$$ Flux=\iint_{R} (\dfrac{\partial M}{\partial x}+\dfrac{\partial N}{\partial y}) \space dx \space dy \\ = \iint_{R} (\dfrac{\partial (2xy+x)}{\partial x}+\dfrac{\partial (xy-y)}{\partial y}) dx \space dy \\=\int_{0}^1 \int_{0}^1(2y+x) dx dy \\=\dfrac{3}{2}$$
The tangential form for Green Theorem can be written as follows:
Counterclockwise Circulation: $\oint_C F \cdot T ds= \iint_{R} (\dfrac{\partial N}{\partial x}-\dfrac{\partial M}{\partial y}) dx dy \\ \iint_{R} (y-2x) \space dy \space dx \\=\int_{0}^1 \int_{0}^1(y-2x) \space dx \space dy \\=\dfrac{-1}{2}$
