<s>[INST] What should be added to $x^4 + 2x^3 - 2x^2 + x - 1$ to make it exactly divisible by $x^2 + 2x - 3$? I'm a ninth grader so please try to explain the answer in simple terms . 
I cant fully understand the explanation in my book .
It just assumes that the expression that should be added has a degree of 1. 
I apologize if this question is too simple or just stupid but this is a genuine doubt. 
 [/INST] Imagine that we add $ax+b$ to the given polynomial, obtaining a new polynomial 
$$P(x)= x^4 +2x^3-2x^2+x-1+ax+b.$$
Note that $x^2+2x-3=(x+3)(x-1)$. So if $x^2+2x-3$ divides our new polynomial $P(x)$, then $P(1)=0$ and $P(-3)=0$. 
We have 
$$P(1)=a+b+1, \qquad\text{and}\qquad P(-3)=-3a+b+5.$$
Solve the system of linear equations $a+b+1=0$, $-3a+b+5=0$ for $a$ and $b$. 
Remark: We have skipped a logical step that in principle should not be skipped. If $ax+b$ is to work, our argument shows that $a$ and $b$ must satisfy the two equations. We have not shown that if $a$ and $b$  satisfy the two equations, then $ax+b$ automatically works. This is in fact true, but requires some theory. 
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