A rectangular solid is changed such that the width and lengt

Ok, so here is what I know:

~Old Volume = New Volume

~(L+1)(W+1)(H-9) = (L*W*H)

~(H-9) = 4w

~width, length of original rectangular are equal

So, from that I get:

(L+1)(W+1)(4w)=(W*W*H)

But in the book, the equation differs from mine in two ways.

For starters, mine is L+1)(W+1)(4w)=(W*W*H) while theirs is (W+1)(W+1)(4w)=(W*W*H)

Also, because (H-9)=4w, they derive H=4w+9 then plug it in so (W+1)(W+1)(4w)=(W*W*4w+9)

But here is my (apparently incorrect) reasoning.

L=W in the old rectangle, so why plug "W" into the new rectangle volume?

if (H-9)=4w, then why do I plug 4w into the new rectangle volume and h=4w-9 into the old rectangle formula? It seems unnecessary to have to plug in 4w for (h-9) then derive h=4w-9 and plug in on the other side.

What's the reasoning!

Thanks!