The real part of the complex number (1+2i)4(1−2i)2(3+2i)(4−6i) is

[(1+2i)(1−2i)]2(1+2i)212−18i+8i+12=25[1−4+4i]24−10i=25[−3+4i]24−10i=[−75+100i][24+10i](24−10i)(24+10i)−1800−750i+2400i−1000576+100RealPart =−2800676

The real part of the complex number (1+2i)4(1−2i)2(3+2i)(4−6i) is

[(1+2i)(1−2i)]2(1+2i)212−18i+8i+12=25[1−4+4i]24−10i=25[−3+4i]24−10i=[−75+100i][24+10i](24−10i)(24+10i)−1800−750i+2400i−1000576+100RealPart =−2800676

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The real part of the complex number $\frac{{(1 + 2 i)}^{4} {(1 - 2 i)}^{2}}{(3 + 2 i) (4 - 6 i)}$ is

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$\frac{- 2800}{576}$

$\frac{- 2800}{676}$

$\frac{2800}{676}$

$\frac{2700}{676}$

answer is B.

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Detailed Solution

$\begin{array}{l} \frac{{[(1 + 2 i) (1 - 2 i)]}^{2} {(1 + 2 i)}^{2}}{12 - 18 i + 8 i + 12} \\ = \frac{25 [1 - 4 + 4 i]}{24 - 10 i} \\ = \frac{25 [- 3 + 4 i]}{24 - 10 i} \\ = \frac{[- 75 + 100 i] [24 + 10 i]}{(24 - 10 i) (24 + 10 i)} \\ \frac{- 1800 - 750 i + 2400 i - 1000}{576 + 100} \\ Re a l P a r t = - \frac{2800}{676} \end{array}$