Determine the domain of the function of two variables. \[ g(x, y)=\frac{4}{3 y+8 x^{2}} \] \[ \{(x, y) \mid y \neq \square\} \]

Step 1 ：The domain of a function is the set of all possible input values that will output real numbers. In this case, we need to find the values of x and y that will make the function g(x, y) output real numbers.

Step 2 ：The function g(x, y) is undefined when the denominator is equal to zero. Therefore, we need to find the values of x and y that make the denominator equal to zero.

Step 3 ：So, we need to solve the equation \(3y + 8x^2 = 0\) for y.

Step 4 ：The solution to the equation is \(y = -\frac{8x^2}{3}\). This means that the function g(x, y) is undefined when \(y = -\frac{8x^2}{3}\).

Step 5 ：Therefore, the domain of the function g(x, y) is the set of all real numbers (x, y) such that \(y \neq -\frac{8x^{2}}{3}\).

Step 6 ：Final Answer: \(\boxed{\{(x, y) \mid y \neq -\frac{8x^{2}}{3}\}}\)