Introduction:
To make the polynomial f(x) exactly divisible by x^2 + 2x - 3, we need to add a term or multiple terms to the polynomial in such a way that the remainder of the division is zero. This can be achieved by finding the quotient and remainder when f(x) is divided by x^2 + 2x - 3.

Step 1: Dividing f(x) by x^2 + 2x - 3:
To determine the quotient and remainder, we perform polynomial long division. The divisor is x^2 + 2x - 3 and the dividend is f(x) = x^4 + 2x^3 - 2x^2 + x - 1.



Step 2: Performing polynomial long division:
We start by dividing the highest degree term of the dividend (x^4) by the highest degree term of the divisor (x^2). This gives us x^2. We then multiply x^2 by the divisor, which gives us x^4 + 2x^3 - 3x^2.



Next, we subtract x^4 + 2x^3 - 2x^2 - (x^4 + 2x^3 - 3x^2), which gives us x^2 + x^2 + x = 2x^2 + x.

We then bring down the next term from the dividend, which is x. We divide 2x^2 + x by x^2 + 2x - 3. This gives us 2x as the quotient.



Next, we subtract x^3 + 2x^2 - 3x - (-x^3 + 4x), which gives us -x^2 + 4x + 3x = -x^2 + 7x.

We bring down the next term from the dividend, which is -1. We divide -x^2 + 7x by x^2 + 2x - 3. This gives us -x as the quotient.