Question

the problem involves algebraic expressions: 4x - 2, 4x + 2, 4x - 2 + (1/3x - 1), 4x + 2 + (1/3x - 1), and the fraction \\(\frac{12x^2 + 2x - 2}{3x - 1}\\).

Explanation:

Step1: Factor the numerator

Factor $12x^2 + 2x - 2$. First factor out a 2:
$12x^2 + 2x - 2 = 2(6x^2 + x - 1)$
Factor the quadratic inside: $6x^2 + x - 1 = (3x - 1)(2x + 1)$
So $12x^2 + 2x - 2 = 2(3x - 1)(2x + 1)$

Step2: Simplify the rational expression

Divide the factored numerator by $3x - 1$:
$\frac{2(3x - 1)(2x + 1)}{3x - 1} = 2(2x + 1)$

Step3: Expand the simplified expression

$2(2x + 1) = 4x + 2$

Answer:

$4x + 2$