<s>[INST] How do I factorize a polynomial $ax^2 + bx + c$ where $a \neq 1$ How do I factorize a polynomial $ax^2 + bx + c$ where $a \neq 1$?
E.g. I know that $6x^2 + 5x + 1$ will factor to $(3x + 1)(2x + 1)$, but is there a recipe or an algorithm that will allow me to factorize this?
 [/INST] The roots of a quadratic polynomial $ax^2+bx+c$ are $x_1=\frac{-b+\sqrt{b^2-4ac}}{2a}$ and $x_2=\frac{-b-\sqrt{b^2-4ac}}{2a}$. Then the factorization is just $a(x-x_1)(x-x_2)$.
In your example, $a=6,b=5,c=1$, so $x_1=\frac{-5+\sqrt{5^2-4\cdot 6\cdot1}}{2\cdot 6}=-\frac13$ and $x_2=\frac{-5-\sqrt{5^2-4\cdot 6\cdot1}}{2\cdot 6}=-\frac12$. The factorization is then $6(x+\frac13)(x+\frac12)$, or $(3x+1)(2x+1)$.
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