To multiply the complex numbers [tex]\((3 - 5i)\)[/tex] and [tex]\((5 - 7i)\)[/tex], you can follow these steps:
1. Distribute the terms: Use the distributive property (FOIL method) to expand the expression:
[tex]\[
(3 - 5i)(5 - 7i) = 3 \cdot 5 + 3 \cdot (-7i) + (-5i) \cdot 5 + (-5i) \cdot (-7i)
\][/tex]
2. Multiply each term:
[tex]\[
3 \cdot 5 = 15
\][/tex]
[tex]\[
3 \cdot (-7i) = -21i
\][/tex]
[tex]\[
(-5i) \cdot 5 = -25i
\][/tex]
[tex]\[
(-5i) \cdot (-7i) = 35i^2
\][/tex]
3. Combine the results:
[tex]\[
(3 - 5i)(5 - 7i) = 15 - 21i - 25i + 35i^2
\][/tex]
4. Simplify the imaginary unit [tex]\(i^2\)[/tex]: Recall that [tex]\(i^2 = -1\)[/tex], therefore:
[tex]\[
35i^2 = 35(-1) = -35
\][/tex]
5. Substitute and combine like terms:
[tex]\[
15 - 21i - 25i - 35 = 15 - 35 - 46i
\][/tex]
[tex]\[
= -20 - 46i
\][/tex]
So, the product of [tex]\((3 - 5i)\)[/tex] and [tex]\((5 - 7i)\)[/tex] is [tex]\(-20 - 46i\)[/tex].
Thus, the correct answer is:
[tex]\[
\boxed{-20 - 46i}
\][/tex]
This corresponds to option B.
