$$\frac{(a+1)}{a+2}\times\frac{a^{2}-a-2}{a^{2}+a}$$
$\frac{\left(a-2\right)\left(a+1\right)}{a\left(a+2\right)}$
$$\frac{a+1}{a+2}\times \frac{\left(a-2\right)\left(a+1\right)}{a\left(a+1\right)}$$
$$\frac{a+1}{a+2}\times \frac{a-2}{a}$$
$$\frac{\left(a+1\right)\left(a-2\right)}{\left(a+2\right)a}$$
$$\frac{a^{2}-a-2}{\left(a+2\right)a}$$
$$\frac{a^{2}-a-2}{a^{2}+2a}$$
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$\frac{a^{2}-a-2}{a\left(a+2\right)}$
