Use Green's Theorem to evaluate $\oint_{C} \vec{F} \cdot d \vec{r}$, where $\vec{F}(x, y)=\left\langle-5 x y+9,-8 x^{2}+4\right\rangle$ and $C$ is the triangle with vertices $(0,0),(8,0)$, and $(8,7)$ oriented counterclockwise.
The integral obtained from from Green's Theorem is
\[
\iint_{D} \square d A
\]
where $D$ is the interior of the triangle.

Step 1 ：We are given the vector field \(\vec{F}(x, y)=\langle-5 x y+9,-8 x^{2}+4\rangle\) and the curve C is the triangle with vertices (0,0),(8,0), and (8,7) oriented counterclockwise.

Step 2 ：We are asked to evaluate the line integral \(\oint_{C} \vec{F} \cdot d \vec{r}\) using Green's Theorem.

Step 3 ：Green's Theorem states that for a positively oriented, piecewise-smooth simple closed curve C in the plane and a vector field F with components having continuous first partial derivatives on an open region containing C, we have: \(\oint_{C} \vec{F} \cdot d \vec{r} = \iint_{D} (curl \vec{F}) \cdot d \vec{A}\), where D is the region bounded by C, and curl F is the curl of the vector field F.

Step 4 ：The curl of a vector field F = in two dimensions is given by \(\partial N/\partial x - \partial M/\partial y\).

Step 5 ：So, we need to find the curl of the given vector field \(\vec{F}(x, y) = \langle-5xy + 9, -8x^2 + 4\rangle\).

Step 6 ：Then, we need to evaluate the double integral of the curl over the region D, which is the interior of the triangle with vertices (0,0), (8,0), and (8,7).

Step 7 ：Let M = -5*x*y + 9 and N = 4 - 8*x**2, then the curl of F is -11*x.

Step 8 ：The limits of integration for x are from 0 to 8 and for y are from 0 to 7.

Step 9 ：Evaluating the double integral, we get -2464.

Step 10 ：Final Answer: The value of the line integral \(\oint_{C} \vec{F} \cdot d \vec{r}\) is \(\boxed{-2464}\).