\(\displaystyle\\
\frac{2}{x+2} + \frac{1}{5} = \frac{6}{x+5}\\\\
\frac{6}{x+5}-\frac{2}{x+2} =\frac{1}{5}\\\\
\frac{6(x+2)}{(x+5)(x+2)}-\frac{2(x+5)}{(x+5)(x+2)} =\frac{1}{5}\\\\
\frac{6(x+2)-2(x+5)}{(x+5)(x+2)}=\frac{1}{5}\\\\
\frac{6x+12-2x-10}{x^2+2x+5x+10}=\frac{1}{5}\\\\
\frac{4x+2}{x^2+7x+10}=\frac{1}{5}\\\\
x^2+7x+10 = 5(4x+2)\\\\
x^2+7x+10 = 20x+10~~~\Big| -10 \\\\
x^2+7x = 20x \\ \\
x^2 +7x-20x =0\\\\
x^2 - 13x = 0\\\\
x(x-13)=0\\\\
\boxed{x_1 =0}\\\\
x-13=0\\\\
\boxed{x_2=13}\)
