Question 10210
Well, you've got the right idea!

A = L*W but L = 2W+8 Because the length (L) is 8 ft more (+8) than twice the width (2W).

{{{(2W+8)(W) = 540}}}
{{{2W^2 + 8W = 540}}} Subtract 540 from both sides.
{{{2W^2 + 8W - 540 = 0}}} Solve this quadratic for W. Factor a 2 to simplify it a bit.

{{{w^2 + 4W - 270 = 0}}}  Since this doesn't factor you can solve it by using the quadratic formula:  {{{W=(-b+-sqrt(b^2-4ac))/2a}}}

Here, a = 1, b = 4, and c = -270.

{{{W = (-4+-sqrt(4^2-4*1*(-270)))/(2*1)}}}

{{{W = (-4+-sqrt(16+1080))/2}}}

{{{W = (-4+-sqrt(1096))/2}}}

{{{W = -2+(1/2)(33.106)}}} or {{{W = -2-(1/2)(33.106)}}}

{{{W = -2+16.55}}} or {{{W = -2-16.55}}}

{{{W = 14.55}}} or {{{W = -18.55}}} Discard the 2nd solution as width can only be positive. Note: {{{sqrt(1096) = 33.1058907145}}} approx, so I rounded it to 33.106 Finally:
 The length = 2W+8 = 2(14.55)+8 = 29.1+8 = 37.1 ft.
The width = 14.55 ft.

Check:

A = L*W = (37.1)(14.55) = 539.805 sq.ft. which is pretty darn close to 540 sq.ft.