Question 198164
<font size = 9 color = "red">Edwin's solution:</b></font>

A Jewelery Box has a length that is 2 inches longer than the width and a height that is 1 inch smaller than the width. It has a volume of 140 cubic inches. What is the width? 
<pre><font size = 4 color = "indigo"><b>
{{{drawing(400,229,-1,13,-1,7, locate(3,0,LENGTH),locate(9,2,WIDTH),

line(0,0,7,0), line(7,0,12,2), line(12,2,12,6), line(12,6,5,6), locate(.2,2,HEIGHT),
line(5,6,0,4), line(0,4,0,0),line(0,4,7,4), line(7,4,12,6), line(7,0,7,4)

)}}}
Let the {{{WIDTH=w}}}
</pre></font></b>
>>...length that is 2 inches longer than the width...<<
<pre><font size = 4 color = "indigo"><b>
So {{{LENGTH = WIDTH + 2}}} or

   {{{LENGTH = w + 2}}}
</pre></font></b>
>>...a height that is 1 inch smaller than the width...<<
<pre><font size = 4 color = "indigo"><b>
So {{{HEIGHT = WIDTH - 1}}} or

   {{{HEIGHT = w - 1}}}

{{{drawing(400,229,-1,13,-1,7, locate(3,0,w+2),locate(9,1.5,w),

line(0,0,7,0), line(7,0,12,2), line(12,2,12,6), line(12,6,5,6), locate(.2,2,w-1), 
line(5,6,0,4), line(0,4,0,0),line(0,4,7,4), line(7,4,12,6), line(7,0,7,4)

)}}}

The formula for volume is

{{{VOLUME=LENGTH*WIDTH*HEIGHT}}}
</pre></font></b>
>>...It has a volume of 140 cubic inches...<<
<pre><font size = 4 color = "indigo"><b>
So we substitute {{{140}}} for {{{VOLUME}}},
{{{w+2}}} for {{{LENGTH}}}, and {{{w-1}}} for {{{HEIGHT}}}:

{{{140=(w+2)*w*(w-1)}}}

First we multiply the {{{w}}} by the {{{(w-1)}}}
by distributing:

{{{140=(w+2)*(w^2-w)}}}

Now we use FOIL on the right:

{{{140=w^3-w^2+2w^2-2w}}} 

{{{140=w^3+w^2-2w}}}

{{{0=w^3+w^2-2w-140}}}

{{{w^3+w^2-2w-140=0}}}

The possible roots of this equation are the
factors of 140, which are

±1, ±2, ±4, ±5, ±10, ±14, ±28, ±35, ±70, ±140  

Start trying those with synthetic division and
sooner or later you'll get around to trying
x=5

5 | 1  1 -2 -140
  |    5 30  140
  --------------
    1  6 28    0

So the polynomial has been factored as

{{{w^3+w^2-2w-140=0}}}

has been factored as

{{{(x-5)(x^2+6x+28)=0}}}

{{{x-5-0}}} so {{{x=5}}}

{{{x^2+6x+28=0}}} just give imaginary solutions.

So the {{{WIDTH = w=5INCHES}}}

Edwin</pre>