\(\displaystyle\frac{{{2}{x}}}{{{x}+{2}}}={x}–{1}\)

\(\displaystyle\frac{{{2}{x}}}{{{x}+{2}}}\times{\left({x}+{2}\right)}={\left({x}–{1}\right)}{\left({x}+{2}\right)}\)

\(\displaystyle{2}{x}={\left({x}–{1}\right)}{\left({x}+{2}\right)}\)

\(\displaystyle{2}{x}={x}^{{2}}–{x}+{2}{x}–{2}–{2}{x}\)

\(\displaystyle{x}^{{2}}–{x}+{2}{x}–{2}–{2}{x}=\)

\(\displaystyle{x}^{{2}}–{x}–{2}={0}\)

\(\displaystyle{x}^{{2}}–{2}{x}+{x}–{2}={0}\)

\(\displaystyle{\left({x}+{1}\right)}{\left({x}–{2}\right)}={0}\)

Thus, x = -1. x = 2.

\(\displaystyle\frac{{{2}{x}}}{{{x}+{2}}}\times{\left({x}+{2}\right)}={\left({x}–{1}\right)}{\left({x}+{2}\right)}\)

\(\displaystyle{2}{x}={\left({x}–{1}\right)}{\left({x}+{2}\right)}\)

\(\displaystyle{2}{x}={x}^{{2}}–{x}+{2}{x}–{2}–{2}{x}\)

\(\displaystyle{x}^{{2}}–{x}+{2}{x}–{2}–{2}{x}=\)

\(\displaystyle{x}^{{2}}–{x}–{2}={0}\)

\(\displaystyle{x}^{{2}}–{2}{x}+{x}–{2}={0}\)

\(\displaystyle{\left({x}+{1}\right)}{\left({x}–{2}\right)}={0}\)

Thus, x = -1. x = 2.