What must be subtracted from 4x4 - 2x3 - 6x2 + x - 5 so that the answer is exactly divisible by x + x - 2

To solve this problem, what we have to do is to divide the whole equation 4 x^4 - 2 x^3 - 6 x^2 + x - 5 with the equation 2 x^2 + x - 1. Whatever remainder we get must be the value that we have to subtract from the main equation 4 x^4 - 2 x^3 - 6 x^2 + x - 5 for it to be exactly divisible by 2 x^2 + x - 1.

By using any method, I used long division we get a remainder of - 6.

Therefore we have to subtract - 6 from the main equation which results in:

4 x^4 - 2 x^3 - 6 x^2 + x - 5 - (-6) = 4 x^4 - 2 x^3 - 6 x^2 + x + 1