Split the second term in 6x^2 + x - 2 into two terms
6x^3 + 4x - 3x - 2/3x^2 + 14x + 8
Factor out the common terms in the first two term, then in the last two terms
2x(3x + 2) - (3x + 2)/3x^2 + 14x + 8
Factor out the common term 3x + 2
(3x + 2)(2x - 1)/3x^2 + 14x + 8
Split the second term in 3x^2 + 14x + 8 into two terms
(3x + 2)(2x - 1)/3x^2 + 12x + 2x + 8
Factor out common terms in the first two terms, then in the last two terms
(3x + 2)(2x - 1)/3x(x + 4) + 2(x + 4)
Factor out the common term x + 4
(3x + 2)(2x - 1)/(x + 4)(3x + 2)
Cancel 3x + 2
2x - 1/x + 4