Factoring multi-variable expressions

Question

I'm asked to simplify the following multivariable expression. $$ \frac{2x^2-3xy-2y^2+3x-6y}{x-2y} $$

The implicit hint here is that if it can be simplified it will invariably involve factoring out $x-2y$ from the numerator and then simplifying giving some removable discontinuity. The thing is I cannot identify factors in the numerator such that I find the factor $x-2y$.

I am not used to factoring multivariable expressions so is there any trick or method I am missing that would help me with this problem and see the factors more easily?

Answer 1

Have you tried long division?

$(2x^2-3xy-2y^2+3x-6y) : (x-2y)=2x+y+3$

$-(2x^2-4xy)$

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$=xy-2y^2+3x-6y$

$-(xy-2y^2)$

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$=3x-6y$

$-(3x-6y)$

---

$=0$

Sorry for the awful notation, but I do not know better, and I think it is clear neverless.

Answer 2

$$2x^2-3xy-2y^2+3x-6y=2x^2-4xy+xy-2y^2+3x-6y=(x-2y)(2x+y+3).$$