To determine which complex number is equivalent to the given expression, follow these steps:
1. Identify the given expression: [tex]\[
(-45 - 22i) + 2(5 - 3i)(5 + 3i)
\][/tex]
2. Simplify the expression inside the parentheses: Start by calculating [tex]\((5 - 3i)(5 + 3i)\)[/tex]. Recognize that this is a product of conjugates. The formula for the product of conjugates is: [tex]\[
(a - bi)(a + bi) = a^2 + b^2
\][/tex] Here, [tex]\(a = 5\)[/tex] and [tex]\(b = 3\)[/tex]: [tex]\[
(5 - 3i)(5 + 3i) = 5^2 + 3^2 = 25 + 9 = 34
\][/tex] So: [tex]\[
(5 - 3i)(5 + 3i) = 34
\][/tex]
3. Multiply the result by 2: [tex]\[
2 \times 34 = 68
\][/tex]
4. Add to the initial complex number [tex]\((-45 - 22i)\)[/tex]: [tex]\[
(-45 - 22i) + 68
\][/tex]
5. Separate the real and imaginary parts: [tex]\[
\text{Real part: } -45 + 68 = 23
\][/tex] [tex]\[
\text{Imaginary part : } -22i \text{ remains.}
\][/tex]
6. Combine these to form the final complex number: [tex]\[
23 - 22i
\][/tex]
Thus, the complex number equivalent to the given expression is: [tex]\[
\boxed{23 - 22i}
\][/tex] So the correct answer is: B. [tex]\(23 - 22i\)[/tex]
