I think you made a mistake, the width of the bigger square is (5+3x), not 8x.
It is possible to not use the Quadratic Formula, just takes some time to do number plug-ins. I did Quadratic Formula cause I do not want to deal with the plug-ins since there is a 2x.
Another way of solving this problem would be:
Big Square - Pathway = Small Square
(5+3x)(5+3x) - 132 = (5+x)(5+x)
25+30x+9x^2 - 132 = 25+10x+x^2
8x^2+20x-132 = 0
2x^2+5x-33 = 0
Try numbers for plug-in so the sum will added up to 5x, this one is actually not very difficult, after I tried it.
(2x+11)(x-3) = 0 [11x - 6x = 5x]
x = -5.5 or 3
(x+5)(x+5) = 8(8) = 64