Answer
$\displaystyle \frac{22x+5}{10\sqrt{(x-5)(4x+3)}}$
Work Step by Step
$\displaystyle \sqrt{4x+3}\cdot\frac{1}{2\sqrt{x-5}}+\sqrt{x-5}\cdot\frac{1}{5\sqrt{4x+3}},\quad x>5$
$=\displaystyle \frac{\sqrt{4x+3}}{2\sqrt{x-5}}+\frac{\sqrt{x-5}}{5\sqrt{4x+3}}$
... LCD =$2\sqrt{x-5}\cdot 5\sqrt{4x+3}$= $10\sqrt{x-5}\sqrt{4x+3}$
$=\displaystyle \frac{\sqrt{4x+3}\cdot 5\cdot\sqrt{4x+3} +\sqrt{x-5}\cdot 2\sqrt{x-5}}{10\sqrt{x-5}\sqrt{4x+3}}$
... apply $\sqrt{a}\cdot\sqrt{a}=\sqrt{a^{2}}=|a|$
$=\displaystyle \frac{5|4x+3|+|x-5|}{10\sqrt{x-5}\sqrt{4x+3}}$
... since $x>5$, both absolute brackets can be replaced with parentheses
$=\displaystyle \frac{5(4x+3)+2(x-5)}{10\sqrt{x-5}\sqrt{4x+3}}$
... numerator: distribute and simplify
... in the denominator, $\sqrt{a}\cdot\sqrt{b}=\sqrt{ab}$
$=\displaystyle \frac{20x+15+2x-10}{10\sqrt{(x-5)(4x+3)}}$
$=\displaystyle \frac{22x+5}{10\sqrt{(x-5)(4x+3)}}$
