Question

What is the quotient of?

Answer 1

You notice that all expressions are quadratic ones.
To be able to simplify, you must find the root (or the zeros) of each expression in applying the following formula :
x' = [-b + √(b²-4ac)]/2a    and x" = [-b - √(b²-4ac)]/2a   

Once you find the roots you will re-write EACH expression under the form:
(x-x')(x-x"). After computing you'll find:

for w²-8w-9 , w' = 9 and w" = -1 → (w-w')(w-w") →(w-9)(w+1) (#1)
for  w²-1 you can expand it as→→→→→→→→→(w-1)(w+1) (#2)
for w²+10w+9,  w' = -9 and w" = -1 → (w-w')(w-w") →(w+9)(w+1) (#3)
for w²-10w+9,  w' = 9 and w" = 1 → (w-w')(w-w") →(w-9)(w-1) (#4)

Now we (#1)/(#2) : (#3)/(#4), which you can re-write (#1)/(#2) x (#4)/(#3)

Replace  (#1)/(#2) x (#4)/(#3), by their respective values:
[(w-9)(w+1)/(w-1)(w+1)] x [(w-9)(w-1)/(w+9)(w+1)]

After simplification you will get : (w-9)/(w+1) (first answer)