https://r2.erweima.ai/mathbot/image/17264982344068diyuqek.jpg

Let's go through the problems one by one, calculating each complex product.

### a. \((3 - 2i)(5 - i)\)
Use the distributive property (FOIL method):
\[
(3 - 2i)(5 - i) = 3 \cdot 5 + 3 \cdot (-i) - 2i \cdot 5 - 2i \cdot (-i).
\]
Calculating each term:
- \(3 \cdot 5 = 15\),
- \(3 \cdot (-i) = -3i\),
- \(-2i \cdot 5 = -10i\),
- \(-2i \cdot (-i) = 2i^2 = 2(-1) = -2\) (since \(i^2 = -1\)).

Combine all terms:
\[
15 - 3i - 10i - 2 = 13 - 13i.
\]
So, the answer is **\(13 - 13i\)**.

### b. \(-5(8 - 2i)\)
Distribute \(-5\) through the parentheses:
\[
-5(8 - 2i) = -5 \cdot 8 - 5 \cdot (-2i) = -40 + 10i.
\]
So, the answer is **\(-40 + 10i\)**.

### c. \((-5 + 3i)(2 - 3i)\)
Use the distributive property:
\[
(-5 + 3i)(2 - 3i) = -5 \cdot 2 + (-5) \cdot (-3i) + 3i \cdot 2 + 3i \cdot (-3i).
\]
Calculating each term:
- \(-5 \cdot 2 = -10\),
- \(-5 \cdot (-3i) = 15i\),
- \(3i \cdot 2 = 6i\),
- \(3i \cdot (-3i) = -9i^2 = 9\) (since \(i^2 = -1\)).

Combine all terms:
\[
-10 + 15i + 6i + 9 = -1 + 21i.
\]
So, the answer is **\(-1 + 21i\)**.

### d. \(8i(8 - 2i)\)
Distribute \(8i\) through the parentheses:
\[
8i(8 - 2i) = 8i \cdot 8 + 8i \cdot (-2i) = 64i - 16i^2.
\]
Since \(i^2 = -1\):
\[
64i - 16(-1) = 64i + 16.
\]
So, the answer is **\(16 + 64i\)**.

### e. \(\left(\frac{3}{2} + i\right)(5i - 2)\)
Use the distributive property:
\[
\left(\frac{3}{2} + i\right)(5i - 2) = \frac{3}{2} \cdot 5i + \frac{3}{2} \cdot (-2) + i \cdot 5i + i \cdot (-2).
\]
Calculating each term:
- \(\frac{3}{2} \cdot 5i = \frac{15i}{2}\),
- \(\frac{3}{2} \cdot (-2) = -3\),
- \(i \cdot 5i = 5i^2 = -5\) (since \(i^2 = -1\)),
- \(i \cdot (-2) = -2i\).

Combine all terms:
\[
\frac{15i}{2} - 3 - 5 - 2i.
\]
Convert \(\frac{15i}{2}\) to a decimal for simplicity: \(7.5i\), then combine:
\[
-8 + 7.5i - 2i = -8 + 5.5i.
\]
So, the answer is **\(-8 + 5.5i\)**.

### f. \((5i - 3)(2i - 2)\)
Use the distributive property:
\[
(5i - 3)(2i - 2) = 5i \cdot 2i + 5i \cdot (-2) - 3 \cdot 2i - 3 \cdot (-2).
\]
Calculating each term:
- \(5i \cdot 2i = 10i^2 = -10\) (since \(i^2 = -1\)),
- \(5i \cdot (-2) = -10i\),
- \(-3 \cdot 2i = -6i\),
- \(-3 \cdot (-2) = 6\).

Combine all terms:
\[
-10 - 10i - 6i + 6 = -4 - 16i.
\]
So, the answer is **\(-4 - 16i\)**.

### g. \(5i(5 - 2i)\)
Distribute \(5i\) through the parentheses:
\[
5i(5 - 2i) = 5i \cdot 5 + 5i \cdot (-2i) = 25i - 10i^2.
\]
Since \(i^2 = -1\):
\[
25i - 10(-1) = 25i + 10.
\]
So, the answer is **\(10 + 25i\)**.

### h. \(-5\left(2 - \frac{3}{2}i\right)\)
Distribute \(-5\) through the parentheses:
\[
-5\left(2 - \frac{3}{2}i\right) = -5 \cdot 2 - 5 \cdot \left(-\frac{3}{2}i\right) = -10 + \frac{15i}{2}.
\]
So, the answer is **\(-10 + \frac{15i}{2}\)** or **\(-10 + 7.5i\)**.

### i. \((5 - 2i)(3 - 2i)\)
Use the distributive property:
\[
(5 - 2i)(3 - 2i) = 5 \cdot 3 + 5 \cdot (-2i) - 2i \cdot 3 - 2i \cdot (-2i).
\]
Calculating each term:
- \(5 \cdot 3 = 15\),
- \(5 \cdot (-2i) = -10i\),
- \(-2i \cdot 3 = -6i\),
- \(-2i \cdot (-2i) = 4i^2 = -4\) (since \(i^2 = -1\)).

Combine all terms:
\[
15 - 10i - 6i - 4 = 11 - 16i.
\]
So, the answer is **\(11 - 16i\)**.

Would you like to go through any of these in more detail or have any questions?

### Follow-up Questions:
1. How do you multiply two complex numbers in polar form?
2. What is the geometric interpretation of multiplying two complex numbers?
3. How does complex conjugation affect multiplication of complex numbers?
4. What are some applications of complex numbers in engineering or physics?
5. How do you convert a complex number from rectangular to polar form?

### Tip:
Always remember that \(i^2 = -1\). This property is crucial when simplifying expressions involving complex numbers.

Calcula los productos (as shown in the image).

Learn how to calculate complex number products using the distributive property. This step-by-step guide solves each problem from basic complex number multiplication to more advanced expressions. Suitable for students in Grades 10-12.

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