To determine which of the given expressions is equivalent to [tex]\( x^2 + 2x + 2 \)[/tex], we will expand and simplify each expression step by step and compare it to [tex]\( x^2 + 2x + 2 \)[/tex].
### Expression 1: [tex]\((x + 1 - i)(x + 1 - i)\)[/tex]
Expand this expression:
[tex]\[
(x + 1 - i)(x + 1 - i) = (x + 1 - i)^2
\][/tex]
Using the binomial expansion:
[tex]\[
= (x + 1)^2 - 2i(x + 1) + i^2
\][/tex]
[tex]\[
= x^2 + 2x + 1 - 2ix - 2i - 1
\][/tex]
[tex]\[
= x^2 + 2x - 2ix - 2i + i^2
\][/tex]
Since [tex]\( i^2 = -1 \)[/tex]:
[tex]\[
= x^2 + 2x - 2ix - 2i - 1
\][/tex]
[tex]\[
= x^2 + 2x - 2ix - 2i - 1
\][/tex]
This expression simplifies to:
[tex]\[
x^2 + 2x - 2ix - 2i - 1 \neq x^2 + 2x + 2
\][/tex]
### Expression 2: [tex]\((x + 2)(x + 1)\)[/tex]
Expand this expression:
[tex]\[
(x + 2)(x + 1) = x(x + 1) + 2(x + 1)
\][/tex]
[tex]\[
= x^2 + x + 2x + 2
\][/tex]
[tex]\[
= x^2 + 3x + 2
\][/tex]
This is not equal to [tex]\( x^2 + 2x + 2 \)[/tex].
### Expression 3: [tex]\((x + 1 - i)(x + 1 + i)\)[/tex]
This expression is a difference of squares:
[tex]\[
(x + 1 - i)(x + 1 + i) = (x + 1)^2 - i^2
\][/tex]
[tex]\[
= x^2 + 2x + 1 - (-1)
\][/tex]
[tex]\[
= x^2 + 2x + 1 + 1
\][/tex]
[tex]\[
= x^2 + 2x + 2
\][/tex]
This is equal to [tex]\( x^2 + 2x + 2 \)[/tex].
### Expression 4: [tex]\((x - 1 + i)(x - 1 - i)\)[/tex]
This expression is also a difference of squares:
[tex]\[
(x - 1 + i)(x - 1 - i) = (x - 1)^2 - i^2
\][/tex]
[tex]\[
= x^2 - 2x + 1 - (-1)
\][/tex]
[tex]\[
= x^2 - 2x + 1 + 1
\][/tex]
[tex]\[
= x^2 - 2x + 2
\][/tex]
This is not equal to [tex]\( x^2 + 2x + 2 \)[/tex].
From the expanded forms, we can see that the expression equivalent to [tex]\( x^2 + 2x + 2 \)[/tex] is:
[tex]\[
(x + 1 - i)(x + 1 + i)
\][/tex]
