How does quadrupling the side lengths of a rectangle affect its area

Well first you find the area of your basic rectangle with equation:
area (a) = length (L) * width (w)

So we got:
a = L * w

Now we find the area of a new rectangle with quadruple side lengths than our (a) rectangle and im going to call this area A:
A = 4L * 4w or A = 4 (L * w)

we want to see how the ratio of A to a or A/a so we can set both equations equal to 1 by dividing the areas to both sides of their respective equations:
1 = (L * w) / a and
1= 4(L * w) / A

since they both equal 1 then the equations have to equal each other:
(L * w) / a = [4 (L * w) / A

cross multiply
A *(L * w) = a *[4 (L * w)]

divide both sides by (L * w)
A = 4a

as you can see by quadrupling the sides by 4, this small rectangles area would also increase by a factor of four