A Rational Cubic with Imaginary Entry Point

Algebra Level pending

Consider the cubic integer polynomial \(h(x)=3x^3+a_2x^2+a_1x+a_0\) given prime \(|a_0|>15\) and where \(h(1)=12\). Find \(a_0\) such that \(h(x)\) has one rational root and two complex roots.

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