3x^2 + 5x - 4 = a(2x - 1)

3x^2 + 5x - 4 = 2ax - a

3x^2 + 5x - 2ax - 4 + a = 0

We need the discriminant to be > 0

(5 - 2a)^2 - 4(3)(-4 + a) > 0

25 - 20a + 4a^2 + 48 - 12a > 0

4a^2 - 32a + 73 > 0

3x^2+5x-4 = 2ax-a

3x^2+x(5-2a)+(a-4) = 0

(5-2a)^2-12(a-4) ≥ 0 for real roots

3x² + 5x - 4 = a(2x-1) = 2ax - a

3x² + (5-2a)x + (a-4) = 0

discriminant = (5-2a)² - 4·3(a-4) = 4a² - 32a + 73

discriminant < 0 when a ∈ (2.5, 5.5)

3x² + 5x - 4 = a(2x - 1) has real roots when a ≤ 2.5 or a ≥ 5.5

It only has real roots when the value inside the square root is >= 0. So the answer is all of the values of a which satisfy this inequality (you must simplify it).