Question

add and simplify
\\(\frac{3}{5x + 2} + \frac{5x}{25x^2 - 4}\\)
a \\(\frac{20x - 6}{25x^2 - 4}\\)
b \\(\frac{20x + 6}{25x^2 - 4}\\)
c \\(\frac{30x - 10}{25x^2 - 4}\\)
d \\(\frac{5x + 3}{25x^2 + 5x - 2}\\)

Explanation:

Step1: Factor the denominator

First, factor \(25x^2 - 4\) using the difference of squares formula \(a^2 - b^2=(a + b)(a - b)\). Here, \(a = 5x\) and \(b = 2\), so \(25x^2 - 4=(5x + 2)(5x - 2)\).

Step2: Find a common denominator

The first fraction has a denominator of \(5x + 2\), and the second has a denominator of \((5x + 2)(5x - 2)\). So the common denominator is \((5x + 2)(5x - 2)=25x^2 - 4\).
Rewrite the first fraction \(\frac{3}{5x + 2}\) with the common denominator: multiply the numerator and denominator by \(5x - 2\), we get \(\frac{3(5x - 2)}{(5x + 2)(5x - 2)}=\frac{15x - 6}{25x^2 - 4}\).

Step3: Add the fractions

Now add the two fractions: \(\frac{15x - 6}{25x^2 - 4}+\frac{5x}{25x^2 - 4}=\frac{15x - 6 + 5x}{25x^2 - 4}\).
Combine like terms in the numerator: \(15x+5x - 6 = 20x - 6\).
So the result is \(\frac{20x - 6}{25x^2 - 4}\).

Answer:

A. \(\frac{20x - 6}{25x^2 - 4}\)