\[(1+2i)x^{3}-2(3+i)x^{2}+(5-4i)x+2a^{2}=0\]

Let \(a\) be a real number, and the equation above have at least one real root (of \(x\)).

What is the value of \(\sum a^{2}\)?

**Details and Assumptions**:

\(i^{2}=-1\).

\(\sum a^{2}\) represents sum of squares of all possible values of \(a\).

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