We have to verify if (1 + i*sqrt 3)^2 + (1 - i*sqrt 3)^2 = -4

(1 + i*sqrt 3)^2 + (1 - i*sqrt 3)^2

=> 1 + i^2*3 + 2*i*sqrt 3 + 1 + i^2*3 - 2*i*sqrt 3

=> 1 - 3 + 2*i*sqrt 3 + 1 - 3 - 2*i*sqrt 3

=> 1 + 1 - 3 - 3

=> -4

**This proves that (1 + i*sqrt 3)^2 + (1 - i*sqrt 3)^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.**