Hard to simplify?

Algebra Level 3

If \(\alpha,\beta\) are the roots of the quadratic equation

\(x^2-\left( 3+2^{\sqrt{\log_23}}-3^{\sqrt{\log_32}} \right)x-2\left( 3^{\log_32}-2^{\log_23} \right) =0\),

then find the value of \(\left( \alpha^2+\alpha\beta+\beta^2 \right) \).

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