Remainders with complex numbers

Question

Let $ f(x) \in C [x] .$
Suppose $ f(-1+i) = 2+5i $ and $ f(-2-i)=-3. $
Determine the remainder of f(x) divided by $(x+1-i)(x+2+i). $

How would i begin with this question, like how would i determine what f(x) is to begin with?

Answer 1

HINT:

Let $\displaystyle f(x)=A(x+1-i)(x+2+i)+B(x+2+i)+C(x+1-i)$

Put $x=-1+i$ and $-2-i$ one by one to find $B,C$

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Alternatively,

let $\displaystyle f(x)=A(x+1-i)(x+2+i)+Bx+C$

Put $x=-1+i$ and $-2-i$ one by one to find $B,C$

In either cases, $B,C$ are arbitrary constants and $A$ is a polynomial