Answer
$x=-\frac{5+4\sqrt{2}}{7},\:x=\frac{4\sqrt{2}-5}{7}$
Work Step by Step
$\frac{x+\frac{2}{x}}{3+\frac{4}{x}}=5x$
$\frac{x^2+2}{3x+4}=5x$
$\frac{x^2+2}{3x+4}\left(3x+4\right)=5x\left(3x+4\right)$
$x^2+2=5x\left(3x+4\right)$
$x^2+2=15x^2+20x$
$-14x^2-20x+2=0$
By quadratic formula:
$x_{1,\:2}=\frac{-\left(-20\right)\pm \sqrt{\left(-20\right)^2-4\left(-14\right)\cdot \:2}}{2\left(-14\right)}$
$x_{1,\:2}=\frac{-\left(-20\right)\pm \:16\sqrt{2}}{2\left(-14\right)}$
$x_1=\frac{-\left(-20\right)+16\sqrt{2}}{2\left(-14\right)},\:x_2=\frac{-\left(-20\right)-16\sqrt{2}}{2\left(-14\right)}$
$x=-\frac{5+4\sqrt{2}}{7},\:x=\frac{4\sqrt{2}-5}{7}$
