Question 702684
Most of the solution process is drawing the picture and forming the equation.

Draw the area picture,
Scroll to see:

{{{drawing( 300, 300, 0, 12, 0, 12, 
  rectangle(1,1,5,5),
  rectangle(5,2.8,10.5,5),
  locate(0.5,3,X),
  locate(3.1,0.7,X),
  locate(7,5,X+5),
  locate(10.7,4.2,X-6),
  locate(3.1,5,X)  
 )}}}


Total carpeting area for the floor is {{{123=X^2+(X+5)*(X-6)}}}
and working from that,...find that....{{{2X^2-X-153=0}}}


Solution to quadratic formula:  {{{X=(1+- sqrt(1-4*2*(-153)))/(2*2)}}}
(Fill in the steps to compute the arithmetic)...
X=9, the length of the room.