Addition, subtraction, multiplication and division of complex numbersHard

Complex Roots Power Summation

If α and β are the roots of the equation 2z² – 3z – 2i = 0, where $i = \sqrt { - 1}$ , then
$16.\operatorname{Re} \left( {\frac{{{\alpha ^{19}} + {\beta ^{19}} + {\alpha ^{11}} + {\beta ^{11}}}}{{{\alpha ^{15}} + {\beta ^{15}}}}} \right).\operatorname{Im} \left( {\frac{{{\alpha ^{19}} + {\beta ^{19}} + {\alpha ^{11}} + {\beta ^{11}}}}{{{\alpha ^{15}} + {\beta ^{15}}}}} \right)$ is equal to

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