We start with the polynomial expression given in the problem: \[ 1 + 12x + 36x^{2} \]
The polynomial can be factored completely. The factorization of the expression is: \[ 1 + 12x + 36x^{2} = \left(6x + 1\right)^{2} \]
The completely factored form of the polynomial is: \[ \boxed{\left(6x + 1\right)^{2}} \]
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