<s>[INST] How to turn $12x^2-8x+1$ into $(2x-1)(6x-1)$ without quadratic equation? After almost seven years I recently started again learning math and have few holes in my algebra knowledge, so I apologize for the beginner question.
My question is:
Is there any simple trick to turn $12x^2-8x+1$ into $(2x-1)(6x-1)$ without using quadratic equation?
And if it's possible, what is step by step procedure?
 [/INST] Write $x=y/12$, so the polynomial becomes
$$
12\frac{y^2}{12^2}-\frac{8y}{12}+1=
\frac{1}{12}(y^2-8y+12)=\frac{1}{12}(y-2)(y-6)
$$
The decomposition of $y^2-8y+12$ is obtained from the search for two numbers whose sum is $8$ and product is $12$.
Substituting back $y=12x$, we have $y-2=2(6x-1)$ and $y-6=6(6x-1)$, so the denominator $12$ cancels out.
Note that in this way the coefficient of $y$ is always $1$. In general, for $a\ne0$, we have, setting $y=x/a$,
$$
ax^2+bx+c=a\frac{y^2}{a^2}+\frac{by}{a}+c=
\frac{1}{a}(y^2+by+ac)
$$
so the trick “sum and product” can be attempted at.
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