Factor \( 6 x^{2}-x+4 \) \( (3 x-4)(x+1) \) \( (3 x+2)(2 x+2) \) \( (6 x-1)(x-4) \) Prime

To factor the quadratic \( 6x^{2}-x+4 \), we can use the discriminant method. The discriminant \( D = b^2 - 4ac \) determines if the quadratic can be factored easily. Here, \( a = 6 \), \( b = -1 \), and \( c = 4 \). Calculating gives \( D = (-1)^2 - 4 \cdot 6 \cdot 4 = 1 - 96 = -95 \). Since the discriminant is negative, the quadratic does not intersect the x-axis and thus cannot be factored into real numbers; it is prime. If you're grappling with factoring in general, remember to always check the discriminant first! A prime polynomial may indicate no rational roots exist, which saves you a lot of guesswork. Always ensure to explore all your options before diving into complex factoring methods!