Why do we factor polynomials the way we do?

Perhaps you're noticing the streetlight effect:

A policeman sees a drunk man searching for something under a streetlight and asks what the drunk has lost. He says he lost his keys and they both look under the streetlight together. After a few minutes the policeman asks if he is sure he lost them here, and the drunk replies, no, and that he lost them in the park. The policeman asks why he is searching here, and the drunk replies, "this is where the light is."

What's the point of factoring a polynomial? It's to undo polynomial multiplication. Our streetlight is our knowledge of polynomial multiplication. How do we get a quadratic that we can factor "nicely" (ie over the integers)? Like this: $$ (ax + b)(cx + d) = acx^2 + (ad + bc)x + bd $$ So if we have a quadratic that we can factor, we had better be able to find four numbers $a,b,c,d$ such that $ac$ is the coefficient of $x^2$, $bd$ is the constant term, and $ad + bc$ is the coefficient of $x$.

That's the ultimate root of the techniques you learned for factoring different types of quadratics over the integers. There are many quadratics that aren't under the streetlamp, so to speak, such as $x^2 - 2$ and $x^2 + 1$; in broadening the circle of light, we meet the real numbers, the quadratic formula, and the complex numbers.

Consider the polynomial $Ax^2+Bx+C$

$$(ax+b)(cx+d)=acx^2+(bc+ad)x+bd$$

Comparing the coefficients, $A=ac, C=bd$ and we want them to satisfies $bc+ad=B$.

Hence we factor $A$ and $C$ and find factors such that $bc+ad=B$.