<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] Since there is no remainder the original equation will be the product of $x-y+1$ with another unknown equation. In order to create the $x^2$, $y^2$, and $xy$ terms we expect the form to be $dx+ey+f$.
$$ax^2+bxy+cy^2+5x−2y+3  = (x−y+1)(dx+ey+f)$$
$$ax^2+bxy+cy^2+5x−2y+3  = dx^2+(e-d)xy-ey^2+(d+f)x+(e-f)y+f$$
working right to left
$$f=3$$
$$e-f=-2, e=1$$
$$d+f=5, d=2$$
$$c=-e, c=-1$$
$$b=e-d, b=-1$$
$$a=d, a=2$$
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