Answer :
[tex] \frac{1}{x} + \frac{1}{2x+2}= \frac{2}{3}\ /\cdot 3x(2x+2) \ \ \ \wedge\ \ \ x \neq 0\ \ \wedge\ \ 2x+2 \neq 0\\ \\.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x\in R-\{0;-1\}\\ \\ \frac{1}{x}\cdot 3x(2x+2) + \frac{1}{2x+2}\cdot 3x(2x+2)= \frac{2}{3}\cdot 3x(2x+2)\\ \\3(2x+2)+3x=2x(2x+2)\\ \\6x+6+3x=4x^2+4x\\ \\-4x^2+9x-4x+6=0\\ \\-4x^2+5x+6=0\ \ \ \Rightarrow\ \ \Delta=5^2-4\cdot(-4)\cdot6=25+96=121\\ \\[/tex]
[tex]x_1= \frac{-5- \sqrt{121} }{2\cdot (-4)}= \frac{-5-11}{-8} = \frac{-16}{-8} =2\ \ \ is\ integer\\ \\x_2= \frac{-5+ \sqrt{121} }{2\cdot (-4)}= \frac{-5+11}{-8} = \frac{6}{-8} =- \frac{3}{4} \ \ \ is\ not\ the\ integer[/tex]
[tex]x_1= \frac{-5- \sqrt{121} }{2\cdot (-4)}= \frac{-5-11}{-8} = \frac{-16}{-8} =2\ \ \ is\ integer\\ \\x_2= \frac{-5+ \sqrt{121} }{2\cdot (-4)}= \frac{-5+11}{-8} = \frac{6}{-8} =- \frac{3}{4} \ \ \ is\ not\ the\ integer[/tex]
