Sorry, this is making me crazy. Thank you for pointing out my error.

w-4/w-9 – w+1/w+9 + w-99/w^2 – 81 =

(w-4)/(w-9) – (w+1)/(w+9)

+ (w-99)/(w^2 – 81)
= [(w-4)(w+9)-(w+1)(w-9)+(w-99)]/[(w-9)(w+9)]
= [w^2+5w -36 -w^2 +8w +9 +w -99]/[(w-9)(w+9)]

= [14w -72]/[(w-9)(w+9)]
= 2(7w -36)/[(w-9)(w+9)]
I don't see any further simplifications

= [w^2+5w -36 -w^2 +8w +9 +w -99]/[(w-9)(w+9)]

= [14w -126]/[(w-9)(w+9)]
= 14(w-9)/[(w-9)(w+9)]
= 14/(w+9)

No problem! Let's break down the expression and simplify it step by step.

The given expression is:
(w-4)/(w-9) - (w+1)/(w+9) + (w-99)/(w^2 - 81)

Step 1: Factorize the denominator of the third term
(w^2 - 81) can be factored as (w-9)(w+9).

Now the expression becomes:
(w-4)/(w-9) - (w+1)/(w+9) + (w-99)/((w-9)(w+9))

Step 2: Find the common denominator
The common denominator in this case is (w-9)(w+9). To combine the fractions, we need to express each fraction with this common denominator.

(w-4) can be multiplied by (w+9)/(w+9)
(w+1) can be multiplied by (w-9)/(w-9)
(w-99) can stay as is because it already has the common denominator.

Now the expression becomes:
[(w-4)(w+9)]/(w-9)(w+9) - [(w+1)(w-9)]/(w-9)(w+9) + (w-99)/((w-9)(w+9))

Step 3: Distribute and combine like terms
Let's distribute and simplify each term:

[(w-4)(w+9)]/(w-9)(w+9) = (w^2 + 5w - 36)/(w^2 - 81)
[(w+1)(w-9)]/(w-9)(w+9) = (w^2 - 8w - 9)/(w^2 - 81)

Now the expression becomes:
(w^2 + 5w - 36)/(w^2 - 81) - (w^2 - 8w - 9)/(w^2 - 81) + (w-99)/((w-9)(w+9))

Subtracting fractions with common denominators, we get:
[(w^2 + 5w - 36) - (w^2 - 8w - 9)]/(w^2 - 81) + (w-99)/((w-9)(w+9))

Simplifying the numerators:
(w^2 + 5w - 36) - (w^2 - 8w - 9) = w^2 + 5w - 36 - w^2 + 8w + 9 = 13w - 27

Now the expression becomes:
(13w - 27)/(w^2 - 81) + (w-99)/((w-9)(w+9))

Step 4: Combine fractions
To combine the fractions, we need a common denominator, which is (w-9)(w+9).

The expression becomes:
[(13w - 27)(w-9) + (w-99)]/((w-9)(w+9))

Simplifying the numerator:
(13w - 27)(w-9) + (w-99) = 13w^2 - 171w + 243 + w - 99 = 13w^2 - 170w + 144

Now the expression simplifies to:
(13w^2 - 170w + 144)/((w-9)(w+9))

This is the final simplified form of the expression: (13w^2 - 170w + 144)/((w-9)(w+9))