To solve this equation, we can first expand and simplify the left side:

(3x+2y-4)^2 = (3x+2y-4)(3x+2y-4)
= 9x^2 + 6xy - 12x + 6xy + 4y^2 - 8y - 12x - 8y + 16
= 9x^2 + 12xy - 24x + 4y^2 - 16y + 16

Now, the equation becomes:

9x^2 + 12xy - 24x + 4y^2 - 16y + 16 + |3x - 5y + 3| = 0

Since we have an absolute value term, we need to consider two cases:

Case 1: 3x - 5y + 3 ≥ 0
In this case, we can remove the absolute value signs, so the equation becomes:

9x^2 + 12xy - 24x + 4y^2 - 16y + 16 + 3x - 5y + 3 = 0

Simplify this to get:

9x^2 + 12xy - 21x + 4y^2 - 21y + 19 = 0

Case 2: 3x - 5y + 3 < 0
In this case, the absolute value sign flips to negative, so the equation becomes:

9x^2 + 12xy - 24x + 4y^2 - 16y + 16 - (3x - 5y + 3) = 0

Simplify this to get:

9x^2 + 12xy - 21x + 4y^2 - 21y + 13 = 0

These are the two possible equations depending on the value of the absolute value term.