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it is given that $$a + \frac{1}{a}$$ = -2
it is possible only when a = -1
hence a + 2 = 1
and so
$$(a+2)^{2}+\frac{1}{(a+2)^{3}}$$ = $$-1^2 + \frac{1}{-1^2}$$ = 2
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