Question 1066615
Let {{{ x }}} = the width of the border 
around the rug. The width and length
both get increased by {{{ 2x }}}
{{{ ( 18 - 2x )*( 31 - 2x ) = 300 }}}
{{{ 558 - 62x - 36x + 4x^2 = 300 }}}
{{{ 4x^2 - 98x + 258 = 0 }}}
{{{ 2x^2 - 49x + 129 = 0 }}}
Use quadratic formula
{{{ x = (-b +- sqrt( b^2 - 4*a*c )) / (2*a) }}}
{{{ a = 2 }}}
{{{ b = -49 }}}
{{{ c = 129 }}}
{{{ x = (-(-49) +- sqrt((-49)^2 - 4*2*129 )) / (2*2) }}}
{{{ x = ( 49 +- sqrt( 2401 - 1032 )) / 4 }}}
{{{ x = ( 49 +- sqrt( 1369 )) / 4 }}}
{{{ x = ( 49 - 37 )/4 }}} ( the negative square root works here )
{{{ x = 12/4 }}}
{{{ x = 3 }}}
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{{{ ( 18 - 2x )*( 31 - 2x ) = 300 }}}
{{{ 18 - 2x = 18 - 2*3 }}}
{{{ 18 - 2x = 18 - 6 }}}
{{{ 18 - 2x = 12 }}}
and
{{{ 31 - 2x = 31 - 2*3 }}}
{{{ 31 - 2x = 25  }}}
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The dimensions of the rug are 12 x 25 
check:
{{{ 12*25 = 300 }}}
OK