To find the maximum value of the expression (4x - 9y), we need to find the maximum value of x and the minimum value of y that satisfy the given relation.

Given relation: x^2 - 9y^2 - 4x + 6y + 4 = 0

Simplifying the equation, we get:
x^2 - 4x - 9y^2 + 6y + 4 = 0

Rearranging the terms, we get:
x^2 - 4x + 4 - 9y^2 + 6y = 0

Factoring the quadratic in x, we get:
(x - 2)^2 - 9y^2 + 6y = 0

Simplifying further, we get:
(x - 2)^2 = 9y^2 - 6y

Taking the square root on both sides, we get:
x - 2 = ±√(9y^2 - 6y)

Now, let's consider the positive square root:
x - 2 = √(9y^2 - 6y)

Squaring both sides, we get:
(x - 2)^2 = 9y^2 - 6y

Expanding, we get:
x^2 - 4x + 4 = 9y^2 - 6y

Rearranging the terms, we get:
x^2 - 4x - 9y^2 + 6y + 4 = 0

Comparing this equation with the given relation, we can see that it is the same equation.

So, the given relation can be written as:
(x - 2)^2 - 9y^2 + 6y = 0

Now, we need to find the maximum value of x and the minimum value of y that satisfy this equation.

- Find the maximum value of x:
Since (x - 2)^2 is always non-negative, for the given equation to hold true, we need to minimize the value of -9y^2 + 6y.

The minimum value of -9y^2 + 6y can be found by completing the square:
-9y^2 + 6y = -9(y^2 - (2/3)y)

Completing the square, we get:
-9(y^2 - (2/3)y + (1/9)) = -9(y - 1/3)^2 + 1/3

Since -9(y - 1/3)^2 is always non-positive, the minimum value of -9y^2 + 6y is 1/3.

Therefore, the maximum value of x is 2.

- Find the minimum value of y:
Since y is squared in the equation, it can take any value.

Therefore, there is no minimum value of y.

Now, let's substitute these values back into the expression (4x - 9y):

(4x - 9y) = (4(2) - 9y) = 8 - 9y

Since y can take any value, the expression (4x - 9y) will be maximum when y is minimum or tends to negative infinity.

So, the maximum value of (4x - 9y) is