What must be added to x^4+2x^3-2x^2+x-1 if x^2+2x-3 is its factor

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Solution

the negative of the remainder to the original polynomial… or, rearrange the polynomial to feature as a factor x2+2x−3x

x4+2x3−2x2+x−1=x4+2x3–3x2+x2+2x−3−x+2=

=x2(x2+2x−3)+(x2+2x−3)−x+2=(x2+1)(x2+2x−3)−x+2

So it looks like if you add x−2 to your original polynomial, it becomes a multiple of x2+2x−3

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