Question 887311
Cardboard dimensions, x and x+6.
Base area becomes {{{(x-2*2)(x+6-2*2)}}}.  A picture would help to see that.
{{{(x-4)(x+2)=BaseArea}}}.


The height is the 2 cm length cut at each corner so that the flaps folded give height of 2 cm.


VOLUME is BaseArea*Height.
{{{highlight_green(x(x-4)(x+2)=182)}}}


Simplify into general form equaling zero.  Look for roots.  It may be a bit of work finding the possible roots to check; but the roots must be positive, and you should try the smaller ones first.  Not TOO small, since you had some described restrictions of "2" and "6".  You need your x to be of sizes which make sense.


{{{x(x^2-2x-8)-182=0}}}
{{{highlight_green(x^3-2x^2-8x-182=0)}}}.


{{{182=2*91=2*7*13}}}-----Very few roots to try.
Try checking for 7, 14, 26,... still try checking 2 or 1, since this may give simpler resulting polynomial to check.   (Rational Roots Theorem)


<s>There must be a mistake in some of the work and discussion I just gave, because</s> I tried the indicated synthetic divisions (most of them) and found no roots yet.  I am assuming the roots must all be rational.  Using the graphing feature in google, the one single root appears to be slightly less than 7.  <b>This root is about 6.9362....</b>