4x^3 + 8x^2 - x - 2 = 0


4x^3 + 8x^2 - x - 2 = 0
4x^2(x + 2) - 1(x + 2) = 0
(4x^2 - 1)(x + 2) = 0
(2x + 1)(2x - 1)(x + 2) = 0


We are given the equation but we are not asked to find the value of x. We are asked to find the value of 2x - 3. To find this value, we need to use the factorized form of the equation.

(2x + 1)(2x - 1)(x + 2) = 0

For the given equation to be zero, either one of the factors on the left-hand side must be zero.

Therefore, we have three cases:

Case 1: 2x + 1 = 0
2x = -1
x = -1/2

Substituting this value of x in 2x - 3, we get:
2x - 3 = 2(-1/2) - 3 = -4

Therefore, in this case, the value of 2x - 3 is -4.

Case 2: 2x - 1 = 0
2x = 1
x = 1/2

Substituting this value of x in 2x - 3, we get:
2x - 3 = 2(1/2) - 3 = -2

Therefore, in this case, the value of 2x - 3 is -2.

Case 3: x + 2 = 0
x = -2

Substituting this value of x in 2x - 3, we get:
2x - 3 = 2(-2) - 3 = -7

Therefore, in this case, the value of 2x - 3 is -7.


From the above three cases, we can conclude that the value of 2x - 3 can be either -4, -2, or -7 depending on the value of x.