Certainly! Let's simplify the expression [tex]\((1 - 5i)(3 + 7i)\)[/tex].
We use the distributive property (also known as the FOIL method for binomials):
[tex]\[
(1 - 5i)(3 + 7i) = 1 \cdot 3 + 1 \cdot 7i + (-5i) \cdot 3 + (-5i) \cdot 7i
\][/tex]
Let's calculate each term separately:
1. The product of the real parts:
[tex]\[
1 \cdot 3 = 3
\][/tex]
2. The product of the real part of the first complex number and the imaginary part of the second:
[tex]\[
1 \cdot 7i = 7i
\][/tex]
3. The product of the imaginary part of the first complex number and the real part of the second:
[tex]\[
-5i \cdot 3 = -15i
\][/tex]
4. The product of the imaginary parts:
[tex]\[
-5i \cdot 7i = -35i^2
\][/tex]
Recall that [tex]\(i^2 = -1\)[/tex]. Therefore:
[tex]\[
-35i^2 = -35 \cdot (-1) = 35
\][/tex]
Now, let's combine all the calculated parts:
[tex]\[
3 + 7i - 15i + 35
\][/tex]
Combine the real parts (3 and 35):
[tex]\[
3 + 35 = 38
\][/tex]
Combine the imaginary parts (7i and -15i):
[tex]\[
7i - 15i = -8i
\][/tex]
Thus, the simplified expression is:
[tex]\[
38 - 8i
\][/tex]
So the correct answer is [tex]\(38 - 8i\)[/tex].
The choice corresponds to:
[tex]\[
\boxed{38 - 8i}
\][/tex]
