We start with the polynomial expression given by: \[ (x^{2}+1)(x+9) \]
The expression can be factored as follows: \[ (x + 9)(x^{2} + 1) \]
The fully factorized form of the polynomial is: \[ \left(x + 9\right) \left(x^{2} + 1\right) \]
\(\boxed{(x + 9)(x^{2} + 1)}\)
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