$(x-1)^{x+2}=(x-1)^{x+6}$
$\Longrightarrow (x-1)^{x+2}-(x-1)^{x+6}=0$
$\Longrightarrow (x-1)^{x+2}.(1-(x-1)^{x+4})=0$
[TEX]\Longrightarrow \left[(x-1)^{x+2}=0\\{1-(x-1)^{x+4} = 0} [/TEX]
[TEX]\Longrightarrow \left[x-1=0\\{(x-1)^{x+4} = 1} [/TEX]
[TEX]\Longrightarrow \left[x=1\\{(x-1)^{x+4} = 1 \Longrightarrow \left[\begin{x-1=1}\\{x-1 = -1}} [/TEX]
[TEX]\Longrightarrow \left[x=1\\ {x=2 \\ x=0} [/TEX]