To solve this equation, we can break it down into cases based on the signs of the absolute values.

Case 1: 2x + 3 ≥ 0 and 2x - 1 ≥ 0
In this case, both absolute values are positive, so we can remove the absolute value signs:
2x + 3 + 2x - 1 = 8/3(x + 1)^2 + 2
4x + 2 = 8/3(x^2 + 2x + 1) + 2
4x + 2 = 8/3x^2 + 16/3x + 8/3 + 2
4x + 2 = 8/3x^2 + 16/3x + 14/3
Multiplying everything by 3 to clear the fractions:
12x + 6 = 8x^2 + 16x + 14
Rearranging and simplifying:
8x^2 + 4x - 8 = 0
2x^2 + x - 2 = 0
Factoring:
(2x - 1)(x + 2) = 0
Solving for x:
2x - 1 = 0 or x + 2 = 0
2x = 1 or x = -2
x = 1/2 or x = -2

Case 2: 2x + 3 ≥ 0 and 2x - 1 < 0
In this case, the first absolute value is positive and the second is negative, so we can remove the absolute value signs and change the sign of the second term:
2x + 3 - (2x - 1) = 8/3(x + 1)^2 + 2
2x + 3 - 2x + 1 = 8/3(x^2 + 2x + 1) + 2
4 = 8/3x^2 + 16/3x + 8/3 + 2
4 = 8/3x^2 + 16/3x + 14/3
Multiplying everything by 3 to clear the fractions:
12 = 8x^2 + 16x + 14
Rearranging and simplifying:
8x^2 + 16x + 2 = 0
4x^2 + 8x + 1 = 0
Using the quadratic formula:
x = (-8 ± √(8^2 - 4(4)(1))) / (2(4))
x = (-8 ± √(64 - 16)) / 8
x = (-8 ± √48) / 8
x = (-8 ± 4√3) / 8
x = -1 ± √3 / 2

Case 3: 2x + 3 < 0 and 2x - 1 ≥ 0
In this case, the first absolute value is negative and the second is positive, so we can remove the absolute value signs and change the sign of the first term:
-(2x + 3) + (2x - 1) = 8/3(x + 1)^2 + 2
-2x - 3 + 2x - 1 = 8/3(x^2 + 2x + 1) + 2
-4 = 8/3x^2 + 16/3x + 8/3 + 2
-4 = 8/3x^2 + 16/3x + 14/3
Multiplying everything by 3 to clear the fractions:
-12 = 8x^2 + 16x + 14
Rearranging and simplifying:
8x^2 + 16x + 26 = 0

Case 4: 2x + 3 < 0 and 2x - 1 < 0
In this case, both absolute values are negative, so we can remove the absolute value signs and change the signs of both terms:
-(2x + 3) - (2x - 1) = 8/3(x + 1)^2 + 2
-2x - 3 - 2x + 1 = 8/3(x^2 + 2x + 1) + 2
-4x - 2 = 8/3(x^2 + 2x + 1) + 2
Multiplying everything by 3 to clear the fractions:
-12x - 6 = 8(x^2 + 2x + 1) + 6
-12x - 6 = 8x^2 + 16x + 8 + 6
-12x - 6 = 8x^2 + 16x + 14
Rearranging and simplifying:
8x^2 + 28x + 20 = 0
2x^2 + 7x + 5 = 0
Factoring:
(x + 1)(2x + 5) = 0
Solving for x:
x + 1 = 0 or 2x + 5 = 0
x = -1 or x = -5/2

So the solutions to the equation are:
x = 1/2, x = -2, x = -1 + √3 / 2, x = -1 - √3 / 2, x = -1, x = -5/2