Introduction:
The coefficient of correlation measures the strength and direction of the linear relationship between two variables. It quantifies how closely the data points in a scatter plot lie to a straight line. In this case, we have a linear equation relating the variables X and Y, and we need to find the coefficient of correlation.

Given equation:
The given equation is 2X + 3Y + 4 = 0. This equation represents a straight line in the XY plane.

Step 1: Rearrange the equation:
To analyze the relationship between X and Y, we need to express Y in terms of X or X in terms of Y. Let's rearrange the equation to solve for Y:

2X + 3Y + 4 = 0
3Y = -2X - 4
Y = (-2/3)X - (4/3)

Step 2: Compare with the standard equation:
The equation Y = mx + c is the standard form of a linear equation, where m represents the slope of the line and c represents the y-intercept. By comparing the given equation with the standard form, we can identify the values of m and c.

Comparing Y = (-2/3)X - (4/3) with Y = mx + c:
m = -2/3 (slope)
c = -4/3 (y-intercept)

Step 3: Coefficient of correlation:
The coefficient of correlation (r) is a value between -1 and +1. It indicates the strength and direction of the linear relationship between X and Y.

To calculate the coefficient of correlation, we can use the formula:

r = √(m^2 / (m^2 + 1))

Plugging in the value of m (-2/3):

r = √((-2/3)^2 / ((-2/3)^2 + 1))
r = √(4/9 / (4/9 + 1))
r = √(4/9 / (13/9))
r = √(4/13)
r ≈ 0.632

Conclusion:
The coefficient of correlation between X and Y, given the equation 2X + 3Y + 4 = 0, is approximately 0.632. This value indicates a moderate positive linear relationship between the variables X and Y.