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: Find roots of quadratic

Solve $x^2 + 2x + 2 = 0$ using quadratic formula $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$, where $a=1, b=2, c=2$.

$$ x=\frac{-2\pm\sqrt{2^2-4(1)(2)}}{2(1)}=\frac{-2\pm\sqrt{4-8}}{2}=\frac{-2\pm\sqrt{-4}}{2}=\frac{-2\pm2i}{2}=-1\pm i $$

Step2: Write factored form

If roots are $r_1$ and $r_2$, quadratic is $(x-r_1)(x-r_2)$. Substitute $r_1=-1+i, r_2=-1-i$.

$$ (x-(-1+i))(x-(-1-i))=(x+1-i)(x+1+i) $$

Step3: Verify by expanding

Multiply the factored form:

$$\begin{align*} (x+1-i)(x+1+i)&=(x+1)^2 - i^2\\ &=x^2+2x+1 - (-1)\\ &=x^2+2x+2 \end{align*}$$

Answer:

B. $(x + 1 - i)(x + 1 + i)$