中一英數(Manipulations of Simple Polynomials)

What can be multiplied to x-2 to give a product containing the term -3x^2 ?
Find the polynomial.
[Note : There are more than one answer.]

答案如下 :
Let P(x)=ax^2+bx+c
P(x)(x-2)
=(ax^3+bx^2+cx)-(2ax^2+2bx+2c)
=ax^3+(b-2a)x^2+(c-2b)x-2c
Any polynomial containing the terms ax^2+bx such that
b-2a=-3 is a solution
Possible solution :
2x^2+x+5
x^2-x-7
x^3-x^2-5x
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麻煩用中文詳細解