Assuming f(x)=6x^3-5x^2-12x-4
Using rational root test:
we know p/q: ±1/1,±1/2,±1/3,±1/6,±2/1,±2/2,±2/3,±2/6,±4/1,±4/2,±4/3,±4/6 are the test candidates.
Since we have to prove 2x+1 is a factor of f(x)
we check x=-1/2,
so f(-1/2) = 6x^3-5x^2-12x-4 = 6(-1/2)^3-5(-1/2)^2-12(-1/2)-4 =0
Since f(-1/2) =0
so, x - (-1/2) = x + 1/2 = (2x +1)/2 is a root.
therefore, 2x+1 is a root of f(x)