Question 964189
Try to work with mostly lower cased letters for the variables.


w, width
L, length  (upper case so not to confuse with any digit 1).
A, area of the rug


You already know, {{{A=L*w}}}.
Translate the description into symbolism.


{{{A=160}}};
{{{wL=160}}};
{{{w=-2+(1/2)L}}}.


What do you have?
Basically two equations in the two unknown variables, w and L.  Use simple substitution.


{{{system(wL=160,w=L/2-2)}}}


{{{(L/2-2)L=160}}}
At this point you should no longer be "racking" your brain, because you now have a quadratic equation in one single variable, L.  You already found this far and then asked for help.  
{{{highlight_green(L^2/2-2L=160)}}}
{{{2(L^2/2-2L)=2*160}}}
{{{L^2-4L=320}}}
{{{highlight_green(L^2-4L-320=0)}}}
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Not even <i>trying</i> to factor that, ....
discriminant is {{{(-4)^2-4*1*(-320)=16+1280=highlight_green(1296=(36)^2)}}}.
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General solution for quadratic equation:
{{{L=(4+- sqrt(36^2))/2}}}
{{{L=(4+- 36)/2}}}
Obviously the PLUS-form makes sense only.
{{{L=(4+36)/2}}}
{{{L=40/2}}}
{{{highlight(L=20)}}}.


Use either the area equation OR the description's formula equation to find w.
{{{wL=160}}}
{{{w=160/L}}}
{{{w=160/20}}}
{{{highlight(w=8)}}}