Logarithmic Inequality

Algebra Level 5

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

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

If 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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