Solution:

Given,

3x + 2y = 12 ...(1)

xy = 6 ...(2)

We need to find the value of 27x³ + 8y³.

Step 1: Express y in terms of x using equation (1):

3x + 2y = 12

2y = 12 - 3x

y = (12 - 3x)/2

Step 2: Substitute the value of y from step 1 in equation (2) and solve for x:

xy = 6

x((12 - 3x)/2) = 6

6x - 3x² = 12

3x² - 6x + 12 = 0

x² - 2x + 4 = 0

Using quadratic formula,

x = (2 ± sqrt(4 - 16))/2

x = 1 ± sqrt(3)i

Step 3: Substitute the values of x and y in 27x³ + 8y³:

27x³ + 8y³

= 27(1 + sqrt(3)i)³ + 8((12 - 3(1 + sqrt(3)i))/2)³

= 27(1 + 3sqrt(3)i + 9(-1) + 27sqrt(3)i(-1 - sqrt(3)i)) + 8(169 - 36sqrt(3)i)

= 27(6sqrt(3)i - 18) + 8(169 - 36sqrt(3)i)

= -486 + 108sqrt(3)i + 1352 - 288sqrt(3)i

= 866 - 180sqrt(3)i

Therefore, the value of 27x³ + 8y³ is 866 - 180sqrt(3)i.