To solve this problem, we need to determine the vertices of triangle $EFG$ given that points $A(-8, -6)$, $B(4, 3)$, and $C(5, -3)$ are the midpoints of triangle $EFG$.

We can solve for the coordinates of the vertices $E$, $F$, and $G$ using the midpoint formula. If we let the coordinates of $E$, $F$, and $G$ be $E(x_1, y_1)$, $F(x_2, y_2)$, and $G(x_3, y_3)$, then we have:

1. $A(-8, -6)$ is the midpoint of $E$ and $F$: $\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) = (-8, -6)$ This gives us: $x_1 + x_2 = -16 \quad \text{and} \quad y_1 + y_2 = -12$

2. $B(4, 3)$ is the midpoint of $F$ and $G$: $\left(\frac{x_2 + x_3}{2}, \frac{y_2 + y_3}{2}\right) = (4, 3)$ This gives us: $x_2 + x_3 = 8 \quad \text{and} \quad y_2 + y_3 = 6$

3. $C(5, -3)$ is the midpoint of $G$ and $E$: $\left(\frac{x_3 + x_1}{2}, \frac{y_3 + y_1}{2}\right) = (5, -3)$ This gives us: $x_3 + x_1 = 10 \quad \text{and} \quad y_3 + y_1 = -6$

Now, we have the following system of equations: