Question

which complex number is equivalent to the given expression?\\((-45 - 22i) + 2(5 - 3i)(5 + 3i)\\)\\(\bigcirc\\) a. \\(-11 + 22i\\)\\(\bigcirc\\) b. \\(-30 - 25i\\)\\(\bigcirc\\) c. \\(-13 + 38i\\)\\(\bigcirc\\) d. \\(23 - 22i\\)

Explanation:

Step1: Expand the product term

Use difference of squares: $(a-b)(a+b)=a^2-b^2$. Here $a=5$, $b=3i$.
$$(5-3i)(5+3i)=5^2-(3i)^2=25-9i^2$$
Since $i^2=-1$, substitute:
$$25-9(-1)=25+9=34$$

Step2: Multiply by the coefficient 2

$$2\times34=68$$

Step3: Add the remaining complex term

$$(-45-22i)+68=(-45+68)-22i$$

Step4: Simplify the real part

$$23-22i$$

Answer:

D. $23 - 22i$