The problem asks us to simplify several expressions involving complex numbers. The operations include multiplication, addition, subtraction, and squaring.

1. Problem Description

The problem asks us to simplify several expressions involving complex numbers. The operations include multiplication, addition, subtraction, and squaring.

2. Solution Steps

a. $(5+2i)(8+6i) = 5(8) + 5(6i) + 2i(8) + 2i(6i) = 40 + 30i + 16i + 12i^2 = 40 + 46i + 12(-1) = 40 + 46i - 12 = 28 + 46i$

b. $(\frac{1}{3} + \frac{2}{5}i) + (\frac{1}{2} + \frac{1}{4}i) = (\frac{1}{3} + \frac{1}{2}) + (\frac{2}{5} + \frac{1}{4})i = (\frac{2}{6} + \frac{3}{6}) + (\frac{8}{20} + \frac{5}{20})i = \frac{5}{6} + \frac{13}{20}i$

c. $(-7-3i) + (-4+4i) = (-7-4) + (-3+4)i = -11 + i$

d. $(4+i\sqrt{3}) + (-6-2i\sqrt{3}) = (4-6) + (\sqrt{3} - 2\sqrt{3})i = -2 - i\sqrt{3}$

e. $(6-7i) - (7-6i) = (6-7) + (-7-(-6))i = -1 + (-7+6)i = -1 -i$

f. $(-1-2i)^2 = (-1-2i)(-1-2i) = (-1)(-1) + (-1)(-2i) + (-2i)(-1) + (-2i)(-2i) = 1 + 2i + 2i + 4i^2 = 1 + 4i + 4(-1) = 1 + 4i - 4 = -3 + 4i$

g. $(-5i)(8i) = -40i^2 = -40(-1) = 40$

3. Final Answer

a. $28 + 46i$

b. $\frac{5}{6} + \frac{13}{20}i$

c. $-11 + i$

d. $-2 - i\sqrt{3}$

e. $-1 - i$

f. $-3 + 4i$

g. $40$