First, recall that dividing by a fraction such as a/b, is the same as multiplying by b/a
(12w^2+36w+27)/(w-2) div (6w^2+3w-9)/(w-2) = (12w^2+36w+27)/(w-2) times (w-2)/(6w^2+3w-9)
The w-2 terms cancel out for w!=2...
= (12w^2+36w+27)*1/(6w^2+3w-9) = (12w^2+36w+27)/(6w^2+3w-9)
To simplify further, we will have to factor the top and bottom. We'll open up with factoring out 3 from the top:
=(3(4w^2+12w+9))/(6w^2+3w-9)
Looking at the top, this factors rather quickly into (2w+3)^2...
= (3(2w+3)^2)/(6w^2+3w-9)
The denominator looks like it will factor into the form of (ax-3)(bx+3). This multiplication would yield abx^2 -3bx+3ax-9 = abx^2 +3(a-b)-9 = 0. From this we can determine that ab=6, a-b = 1 by comparing to our original denominator. Looking at the sets of factors of 6, we'd have the options of (a=1,b=6), (a=2,b=3), (a=3,b=2),(a=6,b=1) . Of these, the only one to satisfy a-b=1 is a=3, b=2. Thus we can now factor our denominator...
(3(2w+3)^2)/(6w^2+3w-9)=(3(2w+3)^2)/((3w-3)(2w+3)
We can cancel a 2w+3 term on top and bottom, giving...
= 3(2w+3)/(3w-3)
Since 3w-3 = 3(w-1), we then get..
=(2w+3)/(w-1)
