a) find P(x) : Q(x)
P(x) = 3x^{30 + 2x^{2} + 5x - 4, Q(x) = x - 3
3x^{3} + 2x^{2} + 5x - 4 : x - 3 = 3x^{2} + 11x + 38
3x^{3} - 9x^{2}
11x^{2} + 5x - 4
11x^{2} - 33x
38x - 4
38x - 114
110
P(x) : Q(x) = 3x^{2} + 11x + 38 + 110/(x - 3)
b) find P(x) : Q(x)
P(x) = x^{4} - 3x^{3} + 2x^{2} + 3x - 2, Q(x) = x^{2} + 2x - 1
x^{4} - 3x^{3} + 2x^{2} + 3x - 2 : x^{2} + 2x - 1 = x^{2} -5x + 13
x^{4} + 2x^{3} - x^{2}
-5x^{3} + 3x^{2} + 3x - 2
-5x^{3} - 10x^{2} + 5x
13x^{2} - 2x - 2
13x^{2} + 26x - 13
-28x + 11
P(x) : Q(x) = x^{2} - 5x + 13 - (28x - 11)/(x^{2} + 2x - 1)