Question

1. \\(\frac{3}{n - 5} + \frac{4}{n + 3}\\)

Explanation:

Step1: Find a common denominator

The denominators are \( n - 5 \) and \( n + 3 \), so the common denominator is \( (n - 5)(n + 3) \).

Step2: Rewrite each fraction with the common denominator

For \( \frac{3}{n - 5} \), multiply numerator and denominator by \( n + 3 \): \( \frac{3(n + 3)}{(n - 5)(n + 3)} \)
For \( \frac{4}{n + 3} \), multiply numerator and denominator by \( n - 5 \): \( \frac{4(n - 5)}{(n - 5)(n + 3)} \)

Step3: Add the fractions

\( \frac{3(n + 3) + 4(n - 5)}{(n - 5)(n + 3)} \)

Step4: Expand and simplify the numerator

Expand \( 3(n + 3) = 3n + 9 \) and \( 4(n - 5) = 4n - 20 \)
Add them: \( 3n + 9 + 4n - 20 = 7n - 11 \)
So the result is \( \frac{7n - 11}{(n - 5)(n + 3)} \) (or expanded denominator \( n^2 - 2n - 15 \))

Answer:

\(\frac{7n - 11}{(n - 5)(n + 3)}\) (or \(\frac{7n - 11}{n^2 - 2n - 15}\))