In a rectangle the longest side is 4 times larger than the smallest. If you add 6 cm to the longest side, the area will increase with 108 cm

Let the smallest side = x......so the longest side = 4x

And we have that

(x)(4x + 6) = 108     simplify

4x^2 + 6x - 108 = 0       factor

(2x +12)(2x - 9) = 0        only the second factor gives a positive answer for x, so we have

2x - 9 = 0        add 9 to both sides

2x = 9             divide both sides by 2

x = 9/2 = 4.5

And 4x + 6 = 4(4.5) + 6 = 18 + 6 = 24

So the sides are 4.5cm and 24cm.