\(\frac{1}{\left(x+1\right)^2\left(x+2\right)}=\frac{a}{x+1}+\frac{b}{\left(x+1\right)^2}+\frac{c}{x+2}\)
\(=\frac{a}{x+1}+\frac{b}{x+1^2}+\frac{c}{x+2}\)
\(=\frac{1}{\left(x+1\right)^2\left(x+2\right)=}=\frac{a}{\left(x+1\right)\left(x+2\right)}+\frac{b}{x+2}+\frac{c}{\left(x+1\right)^2\left(x+2\right)}\)
\(\frac{c}{\left(x+1\right)^2}+\frac{a}{\left(x+1\right)\left(x+2\right)}+\frac{b}{\left(x+2\right)}=1\)
\(=\frac{c}{x^2+2c+x+1}+\frac{a}{x^2+3a\left(x+2a\right)}+\frac{b}{x+2b}=1\)
\(=\frac{\left(c+a\right)}{x^2+\left(2+x+1+\frac{a}{x^2+3ax+2a}+\frac{b}{x+2b}\right)=1}\)
\(=\frac{c+a}{x^2+\left(2c+3a+b\right)}x+2a+2b=0\)
\(\frac{c+a=0}{2c+3b=0}2a+2b=0\)
\(c=b=-a\)
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