Question

what is the sum?
\\(\frac{3}{x^2 - 9} + \frac{5}{x + 3}\\)
\\(\bigcirc\\ \frac{8}{x^2 + x - 6}\\)
\\(\bigcirc\\ \frac{5x - 12}{x - 3}\\)
\\(\bigcirc\\ \frac{-5x}{(x + 3)(x - 3)}\\)
\\(\bigcirc\\ \frac{5x - 12}{(x + 3)(x - 3)}\\)

Explanation:

Step1: Factor the denominator

First, factor \(x^2 - 9\) using the difference of squares formula \(a^2 - b^2=(a + b)(a - b)\). So, \(x^2 - 9=(x + 3)(x - 3)\). The expression becomes \(\frac{3}{(x + 3)(x - 3)}+\frac{5}{x + 3}\).

Step2: Find a common denominator

The common denominator of \((x + 3)(x - 3)\) and \(x + 3\) is \((x + 3)(x - 3)\). Rewrite \(\frac{5}{x + 3}\) with the common denominator: \(\frac{5(x - 3)}{(x + 3)(x - 3)}\).

Step3: Add the fractions

Now 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 and simplify the numerator: \(3+5x - 15 = 5x - 12\). So the result is \(\frac{5x - 12}{(x + 3)(x - 3)}\) (or also can be written as \(\frac{5x - 12}{x^2 - 9}\), but looking at the options, \(\frac{5x - 12}{(x + 3)(x - 3)}\) is one of the options (the fourth one) and also \(\frac{5x - 12}{x - 3}\) is incorrect as the denominator should be \((x + 3)(x - 3)\) or simplified, but wait, wait, no—wait, when we have \(\frac{5x - 12}{(x + 3)(x - 3)}\), but also, let's check the second option: \(\frac{5x - 12}{x - 3}\) is wrong because the denominator after adding is \((x + 3)(x - 3)\), but wait, no—wait, in step 3, when we add, the denominator is \((x + 3)(x - 3)\), but let's re - check the calculation. Wait, \(\frac{3}{(x + 3)(x - 3)}+\frac{5(x - 3)}{(x + 3)(x - 3)}=\frac{3+5x - 15}{(x + 3)(x - 3)}=\frac{5x - 12}{(x + 3)(x - 3)}\), which is the fourth option. Also, if we simplify the denominator of the second option, \(\frac{5x - 12}{x - 3}\) has a wrong denominator. Wait, maybe I made a mistake. Wait, no, let's re - do the step 2. Wait, the common denominator is \((x + 3)(x - 3)\), so \(\frac{5}{x + 3}=\frac{5(x - 3)}{(x + 3)(x - 3)}\), then adding to \(\frac{3}{(x + 3)(x - 3)}\) gives \(\frac{3 + 5x-15}{(x + 3)(x - 3)}=\frac{5x - 12}{(x + 3)(x - 3)}\), which is the fourth option. Also, the second option is \(\frac{5x - 12}{x - 3}\), which is missing the \((x + 3)\) in the denominator, so it's incorrect. The fourth option is \(\frac{5x - 12}{(x + 3)(x - 3)}\), which is correct.

Answer:

D. \(\frac{5x - 12}{(x + 3)(x - 3)}\) (assuming the fourth option is D, if we label the options as A, B, C, D: A. \(\frac{8}{x^2 + x - 6}\), B. \(\frac{5x - 12}{x - 3}\), C. \(\frac{-5x}{(x + 3)(x - 3)}\), D. \(\frac{5x - 12}{(x + 3)(x - 3)}\)) So the correct answer is D. \(\frac{5x - 12}{(x + 3)(x - 3)}\)