\(\frac{1}{x\left(x+1\right)}+\frac{1}{\left(x+1\right)\left(x+2\right)}+\frac{1}{\left(x+2\right)\left(x+3\right)}=\frac{3}{10}\)
\(\Leftrightarrow\frac{1}{x}-\frac{1}{x+1}+\frac{1}{x+1}-\frac{1}{x+2}+\frac{1}{x+2}-\frac{1}{x+3}=\frac{3}{10}\)
\(\Leftrightarrow\frac{1}{x}-\frac{1}{x+3}=\frac{3}{10}\)
\(\Leftrightarrow\frac{\left(x+3\right)-x}{x\left(x+3\right)}=\frac{3}{10}\)
\(\Leftrightarrow\frac{3}{x\left(x+3\right)}=\frac{3}{10}\)
\(\Rightarrow x\left(x+3\right)=10=2.\left(2+3\right)\)
\(\Rightarrow x=2\)
pt <=> \(\frac{1}{x}-\frac{1}{x+1}+\frac{1}{x+1}-\frac{1}{x+2}+\frac{1}{x+2}-\frac{1}{x+3}=\frac{3}{10}\)
\(\Leftrightarrow\frac{1}{x}-\frac{1}{x+3}=\frac{3}{10}\)
\(\Leftrightarrow\frac{3}{x\left(x+3\right)}=\frac{3}{10}\)
\(\Leftrightarrow x^2+3x-10=0\)
\(\Leftrightarrow\left(x-2\right)\left(x+5\right)=0\Leftrightarrow\orbr{\begin{cases}x=2\\x=-5\end{cases}}\)