Factoring Quadratic Trinomials I'm currently doing some homework, but I'm COMPLETELY stuck on one problem. I need to factor the following trinomial:
$$5x^2+7xy+2y^2$$
How can I solve this problem? I have no idea what to do because of the different variables.

By the AC method we can reduce to factoring a monic (leading coefficient $= 1).$
$$\begin{eqnarray} f\, &=&\ \ \: \color{#c00}2y^2\ + 7xy\ \ \ +\ \ \ 5x^2 \\ \Rightarrow\ \color{#c00}2f\, &=&\, (\color{#c00}2y)^2 + 7x(\color{#c00}2y)+ \color{#c00}2\cdot 5\ x^2, \ \ {\rm let}\ \ Y = 2y \\ &=&\quad Y^2 + 7x\ Y + (2x)(5x)\\ &=&\,\ \ (Y + 2x)(Y + 5x) \\ &=&\,\ (2y+2x)(2y+5x)\\ \Rightarrow\ f\, &=&\ \ \ \,  (y\ +\ x)(2y+5x)\end{eqnarray}$$
Remark $\ $ Due to unique factorization, this method will always succeed (see the above link for further details).
