What must be added to the polynomial f(x) = x⁴ + 2x³ - 2x² + x - 1 so that the resulting polynomial is exactly divisible by x² + 2x - 3?

Question: What must be added to the polynomial f(x) = x⁴ + 2x³ - 2x² + x - 1 so that the resulting polynomial is exactly divisible by x² + 2x - 3?

Solution:
x² + 2x - 3 ) x⁴ + 2x³ - 2x² + x - 1( x² + 1
x⁴ + 2x³ - 3x²
________________________
x² + x - 1
x² + 2x - 3
________________________
- x + 2

To get exactly divisible, the remainder must be 0
- x + 2 + k = 0
⇒ k = (x - 2)

Hence, the correct option is 1.