Explanation

Having no real linear factors means the quadratic has no real roots: discriminant k²+4(k−3) = (k+6)(k−2) < 0, so −6 < k < 2 (k = 3 would make it linear with a real root, and is outside this range anyway). Note the inequality must be strict: at k = −6 or k = 2 the discriminant is zero and a real repeated root exists. The keyed option is B.