0≤3k2−3k+2⟹k2−3k+2≥0⟹(k−1)(k−2)≥0 By Quadratic Inequalities, k∈(−∞,1]∪[2,∞) Since k is the integer part of x, from this part k can be any integer.
By Hint 4, 3k2−3k+2<1⟹k2−3k+2<3⟹k2−3k−1<0 By Quadratic Functions, the roots of k2−3k−1=0 are k=23±13 By Quadratic Inequalities, 23−13<k<23+13 Since k is an integer, k∈{0,1,2,3}