Solution: Use $ |z|^2 + |w|^2 = |z + w|^2 - 2 extRe(z \overlinew) $. Compute $ |z + w|^2 = |2 + 4i|^2 = 4 + 16 = 20 $. Let $ z \overlinew = a + bi $, then $ extRe(z \overlinew) = a $. From $ z + w = 2 + 4i $ and $ zw = 13 - 2i $, note $ |z|^2 + |w|^2 = (z + w)(\overlinez + \overlinew) - 2 extRe(z \overlinew) = |2 + 4i|^2 - 2a = 20 - 2a $. Also, $ zw + \overlinezw = 2 extRe(zw) = 26 $, but this path is complex. Alternatively, solve for $ |z|^2 + |w|^2 = |z + w|^2 - 2 extRe(z \overlinew) $. However, using $ |z|^2 + |w|^2 = (z + w)(\overlinez + \overlinew) - 2 extRe(z \overlinew) = |z + w|^2 - 2 extRe(z \overlinew) $. Since $ z \overlinew + \overlinez w = 2 extRe(z \overlinew) $, and $ (z + w)(\overlinez + \overlinew) = |z|^2 + |w|^2 + z \overlinew + \overlinez w = |z|^2 + |w|^2 + 2 extRe(z \overlinew) $, let $ S = |z|^2 + |w|^2 $, then $ 20 = S + 2 extRe(z \overlinew) $. From $ zw = 13 - 2i $, take modulus squared: $ |zw|^2 = 169 + 4 = 173 = |z|^2 |w|^2 $. Let $ |z|^2 = A $, $ |w|^2 = B $, then $ A + B = S $, $ AB = 173 $. Also, $ S = 20 - 2 extRe(z \overlinew) $. This system is complex; instead, assume $ z $ and $ w $ are roots of $ x^2 - (2 + 4i)x + (13 - 2i) = 0 $. Compute discriminant $ D = (2 + 4i)^2 - 4(13 - 2i) = 4 + 16i - 16 - 52 + 8i = -64 + 24i $. This is messy. Alternatively, use $ |z|^2 + |w|^2 = |z + w|^2 + |z - w|^2 - 2|z \overlinew| $, but no. Correct approach: $ |z|^2 + |w|^2 = (z + w)(\overlinez + \overlinew) - 2 extRe(z \overlinew) = 20 - 2 extRe(z \overlinew) $. From $ z + w = 2 + 4i $, $ zw = 13 - 2i $, compute $ z \overlinew + \overlinez w = 2 extRe(z \overlinew) $. But $ (z + w)(\overlinez + \overlinew) = 20 = |z|^2 + |w|^2 + z \overlinew + \overlinez w = S + 2 extRe(z \overlinew) $. Let $ S = |z|^2 + |w|^2 $, $ T = extRe(z \overlinew) $. Then $ S + 2T = 20 $. Also, $ |z \overlinew| = |z||w| $. From $ |z||w| = \sqrt173 $, but $ T = extRe(z \overlinew) $. However, without more info, this is incomplete. Re-evaluate: Use $ |z|^2 + |w|^2 = |z + w|^2 - 2 extRe(z \overlinew) $, and $ extRe(z \overlinew) = extRe(raczww \overlinew \cdot \overlinew^2) $, too complex. Instead, assume $ z $ and $ w $ are conjugates, but $ z + w = 2 + 4i $ implies $ z = a + bi $, $ w = a - bi $, then $ 2a = 2 \Rightarrow a = 1 $, $ 2b = 4i \Rightarrow b = 2 $, but $ zw = a^2 + b^2 = 1 + 4 = 5 - Groen Casting