Question:
What must be subtracted from $\left(x^{4}+2 x^{3}-2 x^{2}+4 x+6\right)$ so that the result is exactly divisible by $\left(x^{2}+2 x-3\right) ?$
Solution:
Dividing $\left(x^{4}+2 x^{3}-2 x^{2}+4 x+6\right)$ by $\left(x^{2}+2 x-3\right)$ using long division method, we have
Here, the remainder obtained is (2x + 9).
Thus, the remainder $(2 x+9)$ must be subtracted from $\left(x^{4}+2 x^{3}-2 x^{2}+4 x+6\right)$ so that the result is exactly divisible by $\left(x^{2}+2 x-3\right)$.