Question

what is the sum?
\\(\frac{3}{x^2 - 9} + \frac{5}{x + 3}\\)
\\(\bigcirc\\) \\(\frac{8}{x^2 + x - 8}\\)
\\(\bigcirc\\) \\(\frac{5x - 12}{x - 3}\\)
\\(\bigcirc\\) \\(\frac{-5x}{(x + 3)(\text{3})}\\) (note: likely a typo, should be \\(\frac{-5x}{(x + 3)(x - 3)}\\) or similar)
\\(\bigcirc\\) \\(\frac{5x - 12}{(x + 3)(x - 3)}\\)

Explanation:

Step1: Factor the denominator

Notice that \(x^2 - 9\) is a difference of squares, so \(x^2 - 9=(x + 3)(x - 3)\). So the first fraction becomes \(\frac{3}{(x + 3)(x - 3)}\).

Step2: Find a common denominator

The second fraction is \(\frac{5}{x + 3}\), to get a common denominator of \((x + 3)(x - 3)\), we multiply the numerator and denominator of the second fraction by \((x - 3)\). So \(\frac{5}{x + 3}=\frac{5(x - 3)}{(x + 3)(x - 3)}\).

Step3: Add the fractions

Now we add the two fractions: \(\frac{3}{(x + 3)(x - 3)}+\frac{5(x - 3)}{(x + 3)(x - 3)}=\frac{3 + 5(x - 3)}{(x + 3)(x - 3)}\).

Step4: Simplify the numerator

Expand the numerator: \(3+5(x - 3)=3 + 5x-15 = 5x-12\). So the sum is \(\frac{5x - 12}{(x + 3)(x - 3)}\).

Answer:

\(\frac{5x - 12}{(x + 3)(x - 3)}\) (corresponding to the option: \(\frac{5x - 12}{(x + 3)(x - 3)}\))