To simplify the expression [tex]$(1 - 5i)(3 + 7i)$[/tex], we can use the distributive property (also known as the FOIL method for binomials) to multiply the terms. Here are the steps:
1. Multiply the real parts:
- Multiply the real part of the first complex number by the real part of the second complex number:
[tex]\[
1 \cdot 3 = 3
\][/tex]
2. Multiply the real part by the imaginary part:
- Multiply the real part of the first number by the imaginary part of the second number:
[tex]\[
1 \cdot 7i = 7i
\][/tex]
3. Multiply the imaginary part by the real part:
- Multiply the imaginary part of the first number by the real part of the second number:
[tex]\[
-5i \cdot 3 = -15i
\][/tex]
4. Multiply the imaginary parts:
- Multiply the imaginary part of the first number by the imaginary part of the second number:
[tex]\[
-5i \cdot 7i = -35i^2
\][/tex]
- Since [tex]\(i^2 = -1\)[/tex], we substitute and simplify:
[tex]\[
-35i^2 = -35(-1) = 35
\][/tex]
5. Combine the real parts:
- Add the results of the real parts:
[tex]\[
3 + 35 = 38
\][/tex]
6. Combine the imaginary parts:
- Add the results of the imaginary parts:
[tex]\[
7i - 15i = -8i
\][/tex]
Therefore, the simplified expression is:
[tex]\[
38 - 8i
\][/tex]
So, the correct answer is:
\[
38 - 8i
\
