Solve the equation. Check for extraneous solutions. $\frac{10}{x+2}-\frac{1}{x}=\frac{1}{5x}$

A) $x=\frac{6}{23}$
B) $x=\frac{1}{4}$
C) $x=\frac{3}{14}$
D) $x=\frac{3}{11}$
E) $x=\frac{2}{7}$

Multiplying both sides of the equation by $5x(x+2)$ and simplifying gives us $50x=23x^2+46x+20$, which simplifies to $23x^2+4x-20=0$. We can factor this equation to get $(x-2)(23x+10)=0$, so $x=2$ or $x=-\frac{10}{23}$. However, $x=2$ is an extraneous solution since it makes the denominator of the original equation equal to $0$. Thus, the only valid solution is $x=-\frac{10}{23}$, which corresponds to choice D.