Question 1163225

What is the length of the edge of a cube if its volume could be doubled by an increase of 6 centimeters in one edge, an increase of 12 centimeters in a second edge, and a decrease of 4 centimeters in the third edge?'

I've seen the answer of this question on this website. However, I wish to understand the concepts. Is it alright if a list of concepts may be provided? I'm fine with understanding it on my own, but I don't know where to start. Thank you!
<pre>Let length of one of the edges, be L
Then original volume = L<sup>3</sup>
We then get: {{{matrix(7,3, (L + 6)(L + 12)(L  -  4), "=", 2L^3, (L + 6)(L^2 + 8L  -  48), "=", 2L^3, L(L^2 + 8L  -  48) + 6(L^2 + 8L  -  48), "=", 2L^3, L^3 + 8L^2  -  48L + 6L^2 + 48L - 288, "=", 2L^3, L^3 + 14L^2 - 288, "=", 2L^3, 0, "=", 2L^3 - L^3  -  14L^2 + 288, 0, "=", L^3 - 14L^2 + 288)}}}
Using the Rational Root Theorem (this I prefer), or Synthetic Division, we get one of the roots of the above equation as 6, or the factor, L  -  6.
Using the root, 6, a and the Rational Root Theorem again, or Synthetic Division, we get another root, 12.
Or, using the factor, L - 6 and Long Division of Polynomials, we get the resulting quadratic: {{{L^2 - 8L - 48}}}, which factors to (L - 12)(L + 4),
where we see the other root, 12. 
Therefore, we can say that: {{{matrix(1,7, L^3 - 14L^2 + 288, "=", 0, "=====>", (L  -  6)(L  -  12)(L + 4), "=", 0)}}}
This means that the length of the edge of the cube, or {{{highlight_green(matrix(1,5, L, is, highlight(matrix(1,2, 6, cm)), or, highlight(matrix(1,2, 12, cm))))}}}