Verify the equality (1+i*square root 3)^2 + (1-i*square root3)^2=-4

First, we'll have to expand the binomials and we'll use
the identities:

(a+b)^2 = a^2 + 2ab +
b^2

(a-b)^2 = a^2 - 2ab +
b^2

According to these, we'll
get:

(1+i*sqrt3)^2 = 1 + 2i*sqrt3 - 3 (we've replaced i^2
by -1)

(1+i*sqrt3)^2 = -2 + 2i*sqrt3
(1)

(1-i*sqrt3)^2 = 1 - 2i*sqrt3 -
3

(1-i*sqrt3)^2 = -2 - 2i*sqrt3
(2)

We'll add (1) and (2) and we'll
get:

(1+i*sqrt3)^2+(1-i*sqrt3)^2 = -2 + 2i*sqrt3 -2 -
2i*sqrt3

We'll eliminate like
terms:

(1+i*sqrt3)^2+(1-i*sqrt3)^2 =
-4

We notice that the equality
(1+i*sqrt3)^2+(1-i*sqrt3)^2 = -4 is verified.