Question

which expression is equivalent to $x^2 + 2x + 2$?
$(x + 1 - i)(x + 1 - i)$ $(x - 1 + i)(x - 1 - i)$ $(x + 1 - i)(x + 1 + i)$ $(x + 2)(x + 1)$

Explanation:

Step1: Recall the formula for \((a + b)(a - b)\)

The formula is \(a^2 - b^2\). We can also complete the square for the quadratic expression \(x^2 + 2x + 2\). First, complete the square: \(x^2+2x + 1+1=(x + 1)^2+1\). Since \(i^2=- 1\), we can rewrite \(1\) as \(-i^2\), so \((x + 1)^2+1=(x + 1)^2-(- 1)=(x + 1)^2 - i^2\).

Step2: Apply the difference of squares formula

Now, using the difference of squares formula \(a^2 - b^2=(a - b)(a + b)\), where \(a=(x + 1)\) and \(b = i\). So \((x + 1)^2 - i^2=(x + 1 - i)(x + 1 + i)\).
We can also expand each option:

• Option 1: \((x + 1 - i)(x + 1 - i)=(x + 1)^2-2i(x + 1)+i^2=x^2 + 2x+1-2ix - 2i - 1=x^2+2x-2ix - 2i

eq x^2 + 2x + 2\)

• Option 2: \((x - 1 + i)(x - 1 - i)=(x - 1)^2-i^2=x^2-2x + 1+1=x^2-2x + 2

eq x^2 + 2x + 2\)

• Option 3: \((x + 1 - i)(x + 1 + i)=(x + 1)^2-i^2=x^2+2x + 1+1=x^2+2x + 2\)
• Option 4: \((x + 2)(x + 1)=x^2+3x + 2

eq x^2 + 2x + 2\)

Answer:

C. \((x + 1 - i)(x + 1 + i)\)