x2 + 2x - 3
= x2 + 3x - x - 3
= x(x + 3) - 1 (x + 3)
= (x + 3) (x - 1)
Let p(x) = x4 + 2x3 - 2x2 + 2x - 3
We see that
p(-3) = (-3)4 + 2(-3)3 - 2(-3)2 + 2(-3) - 3
= 81 - 54 - 18 - 6 - 3
= 0
Hence by converse of factor theorem, (x + 3) is a factor of p(x).
Also, we see that
p(1) = (1)4 + 2(1)3 - 2(1)2 + 2(1) - 3
= 0
Hence by converse of factor theorem, (x - 1) is a factor of p(x).
From above, we see that
(x + 3) (x - 1), i.e., x2 + 2x - 3 is a factor of p(x)
⇒ p(x) is exactly divisible by (x2 + 2x - 3).
