Logarithmic Inequality

Algebra Level 5

Find all values of the parameter $$a \in \mathbb{R}$$ for which the following inequality is valid for all $$x \in \mathbb{R}$$.

$1+\log _{ 5 }{ { (x }^{ 2 }+1) } \ge \log _{ 5 }{ (a{ x }^{ 2 }+4x+a) }$

The range of values of a can be expressed in the form of $$(A,B]$$. Then find the value of $$A + B$$.

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