<s>[INST] If $ax^2 +bxy+cy^2+5x-2y+3$ divided by $x-y+1$ has remainder $0$, determine $a$, $b$, and $c$. 
If $ax^2 +bxy+cy^2+5x-2y+3$ divided by $x-y+1$ has remainder $0$, determine $a$, $b$, and $c$. 

I do not know how to approach this problem and would appreciate advice how to proceed. 
 [/INST] In $ax^2+bxy+cy^2+5x−2y+3  = (x−y+1)(dx+ey+f)$, put $x=0, y=1$ to get $c+1 = 0$ and hence $c=-1$. Put $y=0, x=-1$ to get $a-2=0$ and hence $a=2$. Putting $x=1, y = 2$, we get $a+2b+4c+5-4+3 = 0$ and hence $b=-1$.
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