<s>[INST] How to factor the quadratic polynomial $2x^2-5xy-y^2$? How do I factor this polynomial:  $2x^2-5xy-y^2$ ?
 [/INST] To explain how you get from your polynomial to a more traditional quadratic equation: pull out a factor of $y^2$ to get $y^2\cdot\left(2\frac{x^2}{y^2}-5\frac{xy}{y^2}-\frac{y^2}{y^2}\right)$, or equivalently $y^2\cdot\left(2\frac{x^2}{y^2}-5\frac{x}{y}-1\right)$.  Now, setting $t=\frac{x}{y}$, the latter is obviously $y^2\cdot\left(2t^2-5t-1\right)$.  The quadratic can be broken up into two factors by solving the quadratic equation as other answers have mentioned: $2t^2-5t-1 = (at+b)(ct+d)$.  Then plugging this expression back in, we get the original function as $y^2\cdot(at+b)(ct+d)$, or $\left(y\cdot(at+b)\right)\left(y\cdot(ct+d)\right)$; plugging in $x/y$ for $t$ and multiplying through by the factors of $y$ then gives the factorization as $(ax+by)(cx+dy)$.
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