To solve the problem of finding the absolute extrema of the function $f(x, y) = -5x^2 - 3xy + 4y^2 + 5x - 3y - 4$ on the domain $-2 \leq x \leq 2$ and $-3 \leq y \leq 3$, we need to follow a systematic process that involves:

Step 1: Finding the Critical Points

1. First-order partial derivatives: $f_x = \frac{\partial}{\partial x}\left(-5x^2 - 3xy + 4y^2 + 5x - 3y - 4\right) = -10x - 3y + 5$ $f_y = \frac{\partial}{\partial y}\left(-5x^2 - 3xy + 4y^2 + 5x - 3y - 4\right) = -3x + 8y - 3$

2. Set both partial derivatives to zero: $-10x - 3y + 5 = 0 \quad \text{(Equation 1)}$ $-3x + 8y - 3 = 0 \quad \text{(Equation 2)}$

   Solving this system of linear equations will yield the critical points.

Step 2: Solving the System of Equations

From Equation 1: $-10x - 3y + 5 = 0 \quad \Rightarrow \quad y = \frac{-10x + 5}{3}$

Substitute this into Equation 2: $-3x + 8\left(\frac{-10x + 5}{3}\right) - 3 = 0$ Simplifying: $-3x + \frac{-80x + 40}{3} - 3 = 0$ Multiply the whole equation by 3 to eliminate the fraction: $-9x - 80x + 40 - 9 = 0 \quad \Rightarrow \quad -89x + 31 = 0$ $x = \frac{31}{89}$

Substitute $x = \frac{31}{89}$ into the expression for $y$: $y = \frac{-10\left(\frac{31}{89}\right) + 5}{3} = \frac{-310 + 445}{267} = \frac{135}{267} = \frac{5}{9}$

Thus, the critical point is approximately $\left( \frac{31}{89}, \frac{5}{9} \right)$.

Step 3: Evaluate the Function at Critical Points and Boundaries

1. Evaluate at the critical point $\left( \frac{31}{89}, \frac{5}{9} \right)$.

2. Evaluate at the boundaries: The boundaries are given by the constraints $x = -2, 2$ and $y = -3, 3$, and all combinations of these values. We also need to evaluate the function along the edges of the rectangular domain (e.g., where $x$ is fixed and $y$ varies or vice versa).

   For instance, along $x = 2$, evaluate $f(2, y)$ as $y$ varies between $-3$ and $3$. Similarly, do this for other boundary values.

Step 4: Compare Values

Once the function has been evaluated at the critical point and along the boundaries, compare the values to find the absolute maximum and minimum.

Would you like to go through the detailed evaluations of the boundary points and critical point, or should I proceed with just the final extrema values?