z^2 - z = 5 + 5i

z^2 - z + 1/4 = 5 + 5i + 1/4

(z - 1/2)^2 = 21/4 + 5i

z - 1/2 = ±√(21/4 + 5i)

z = 1/2 ± (5/2 + i)

z1 = 1/2 + 5/2 + i = 3 + i

z2 = 1/2 - 5/2 - i = - 2 - i

Product of roots :

(3 + i)(- 2 - i) = - 5 - 5i

The real part of the product of the roots is - 5