Question 1009529
If x is the length of each side of square to remove, then  the area for the base of the box is  {{{(12-2x)(16-2x)}}} and then the volume is  {{{x(12-2x)(16-2x)<=540}}}, according to the inequality for the description.  The flaps created, which are folded up to form the box, is the height, x.


{{{x(12-2x)(16-2x)-540<=0}}}, importantly, came from the first expression being LESS THAN OR EQUAL TO the given volume limit of 540 cubic inches.


{{{2*2x(6-x)(8-x)-4*135<=0}}}

Divide both members by 4,

{{{x(6-x)(8-x)-135<=0}}}

{{{x(48-14x+x^2)-135<=0}}}

{{{highlight_green(x^3-14x^2+48x-135<=0)}}}------Rational Roots Theorem and synthetic division if you want , but graphing tool shows  ONE root of about x=10.7, which is not useful; not enough size on either dimension of the cardboard to cut that size square.   Any possible value for x which could be made MUST BE LESS THAN  6 inches, and based on what is seen on a graph  (using graphing tool), any value x at 6 or less, will satisfy the inequality.