You don't know how to compare the real and imaginary parts of the two sides?
two complex numbers:
a+bi and c+di are equal if and only if: a = c, and b = d.
so if a^{2}+b^{2}+ a + bi = 10 - 2i, then:
a^{2} + b^{2} + a = 10, and b = -2.
since b = -2, a^{2} + b^{2} + a = a^{2} + 4 + a.
since we know this already equals 10, we get:
a^{2} + a - 6 = 0
and a^{2} + a - 6 = (a - 2)(a + 3), so a = 2, or -3.
as a final check, we verify that:
z = 2 - 2i
z = -3 - 2i both satisfy:
z + |z|^{2} = 10 - 2i
2 - 2i + |2 - 2i|^{2} = 2 - 2i + 4 + 4 = (2 + 4 + 4) - 2i = 10 - 2i
-3 - 2i + |-3 - 2i|^{2} = -3 - 2i + 9 + 4 = (-3 + 9 + 4) - 2i = 10 - 2i (checked).