Final answer:
To add the given fractions, simplify the expressions using a common denominator. Then, combine like terms and reduce the expression to lowest terms.
Step-by-step explanation:
To add the given fractions, simplify the expressions first. The common denominator for both fractions is (6w-7)(7-6w). Multiplying the numerators of each fraction by the denominator of the other, we get:
(44w)(7-6w) + (8w+42)(6w-7)
Expanding and collecting like terms, we get:
308w - 264w^2 + 48w^2 - 56w + 336
Combining the like terms, we have:
-216w^2 + 252w + 336
Next, we reduce the expression to lowest terms by dividing every term by the greatest common factor, which is 12. This gives us the final answer:
-18w^2 + 21w + 28