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If $\omega$ is a complex cube root of unity, then $225+\left(3 \omega+8 \omega^2\right)^2+\left(3 \omega^2+8 \omega\right)^2$ is equal to:
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The correct answer is:
248
$225+\left(3 \omega+8 \omega^2\right)^2+\left(3 \omega^2+8 \omega\right)^2$
$=225+9 \omega^2+64 \omega^4+48 \omega^3+9 \omega^4+64 \omega^2+48 \omega^3$
$=225+9 \omega^2+64 \omega+48+9 \omega+64 \omega^2+48$
$=225+73\left(\omega^2+\omega\right)+96$
$=225-73+96=225+23=248$
$=225+9 \omega^2+64 \omega^4+48 \omega^3+9 \omega^4+64 \omega^2+48 \omega^3$
$=225+9 \omega^2+64 \omega+48+9 \omega+64 \omega^2+48$
$=225+73\left(\omega^2+\omega\right)+96$
$=225-73+96=225+23=248$
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