Sure! Let's simplify the expression [tex]\((3 - 7i)(2 - 5i)\)[/tex] step-by-step using the distributive property, often known as the FOIL (First, Outer, Inner, Last) method for binomials.
Given: [tex]\( (3 - 7i)(2 - 5i) \)[/tex]
Breaking it down using FOIL:
1. First (F): Multiply the first terms in each binomial:
[tex]\[
3 \cdot 2 = 6
\][/tex]
2. Outer (O): Multiply the outer terms in the product:
[tex]\[
3 \cdot (-5i) = -15i
\][/tex]
3. Inner (I): Multiply the inner terms in the product:
[tex]\[
(-7i) \cdot 2 = -14i
\][/tex]
4. Last (L): Multiply the last terms in each binomial:
[tex]\[
(-7i) \cdot (-5i) = 35i^2
\][/tex]
Combine all these results:
[tex]\[
6 - 15i - 14i + 35i^2
\][/tex]
Next, remember that [tex]\(i^2 = -1\)[/tex], so we substitute [tex]\(i^2\)[/tex] with [tex]\(-1\)[/tex]:
[tex]\[
35i^2 = 35(-1) = -35
\][/tex]
Now, our expression becomes:
[tex]\[
6 - 15i - 14i - 35
\][/tex]
Combine the like terms (real parts and imaginary parts separately):
Real part: [tex]\(6 - 35 = -29\)[/tex]
Imaginary part: [tex]\(-15i - 14i = -29i\)[/tex]
So, the expression simplifies to:
[tex]\[
-29 - 29i
\][/tex]
Thus, the correct choice is:
[tex]\(-29 - 29i\)[/tex]
So, the answer is:
[tex]\[
\boxed{-29 - 29i}
\][/tex]
