I've began practicing factoring expressions of the form $Ax^2+Bx+C$. One method I've learned is decomposition where B can be split into two terms say, D and E where DE=AC. which results in $Ax^2+Dx+Ex+C$. The $Ax^2+Dx$ and $Ex+C$ are then factored, leaving a common factor which with further factorizing, leads to having the original expression factorized.
How might this method be derived? What line of reasoning led to it? I'm really trying to get a more intuitive understanding of why this method works but don't have much algebra knowledge beyond high school.
Hint $$Ax^2+Dx=\frac{1}{E}\left(AEx^2+DEx\right)=\frac{1}{E}\left(AEx^2+ACx\right)=\frac{Ax}{E}\left(Ex+C\right)$$
and then
$$(Ax^2+Dx)+(Ex+C)=(Ex+C)\left(\frac{Ax}{E}+1\right)$$
also,
$$Ex+C=\frac{1}{D}(DEx+DC)=\frac{1}{D}(ACx+DC)=\frac{C}{D}(Ax+D)$$
and you can get $D$ and $E$ solving:
$$D+E=B\\ D\cdot E=A\cdot C$$