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 Denominators

Factor \(5x + 2\) (wait, original first denominator is \(5x + 2\), second is \(25x^2 - 4=(5x - 2)(5x + 2)\) (difference of squares: \(a^2 - b^2=(a - b)(a + b)\), here \(a = 5x\), \(b = 2\))
So denominators: \(5x + 2\) and \((5x - 2)(5x + 2)\)

Step2: Find Common Denominator

Common denominator is \((5x - 2)(5x + 2)=25x^2 - 4\)

Step3: Rewrite First Fraction

Rewrite \(\frac{3}{5x + 2}\) with common denominator: multiply numerator and denominator by \((5x - 2)\)
\(\frac{3(5x - 2)}{(5x + 2)(5x - 2)}=\frac{15x - 6}{25x^2 - 4}\)

Step4: Add the Fractions

Now add \(\frac{15x - 6}{25x^2 - 4}+\frac{5x}{25x^2 - 4}\) (since second fraction already has common denominator)
Combine numerators: \((15x - 6)+5x = 20x - 6\)
So the sum is \(\frac{20x - 6}{25x^2 - 4}\)

Answer:

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