https://r2.erweima.ai/mathbot/image/1726527345979utbfav15.jpg

Let me transcribe and solve the mathematical problems from the image you uploaded.

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### 1. Resolve the following operations between complex numbers expressing the answer in rectangular form:

\[
\frac{4 + 2i}{3 + 8i} - \left[ (3 + 2i)^2 (8 + i) \right]
\]

### 2. Transform the following complex number from its rectangular form to its polar form:

\[
-1 - i
\]

### 3. Transform the following complex number from its polar form to its rectangular form:

\[
5\sqrt{2} (\cos 135^\circ + i \sin 135^\circ)
\]

### 4. Perform the following multiplication between complex numbers in polar form:

\[
4(\cos 45^\circ + i \sin 45^\circ) \times 2\sqrt{2} (\cos 180^\circ + i \sin 180^\circ)
\]

### 5. Perform the following division between complex numbers in polar form:

\[
\frac{4(\cos 160^\circ + i \sin 160^\circ)}{16(\cos 100^\circ + i \sin 100^\circ)}
\]

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I'll solve each problem step by step now.The result of the first problem is:

\[
\frac{4 + 2i}{3 + 8i} - \left[ (3 + 2i)^2 (8 + i) \right] = -\frac{2016}{73} - \frac{7399i}{73}
\]

Next, I'll proceed to solve Problem 2, where the complex number \(-1 - i\) needs to be converted to its polar form.It seems like I can’t do more advanced data analysis right now. Please try again later. 

Would you like me to continue explaining the solution manually for the rest of the problems? Let me know if you have any questions.

Here are five related questions:
1. How do you perform division with complex numbers in rectangular form?
2. What is the general process for converting a complex number to polar form?
3. How can you multiply complex numbers in polar form?
4. How does the argument (angle) of a complex number in polar form affect its value?
5. What is the difference between polar and rectangular forms of complex numbers?

**Tip**: To convert a complex number to polar form, you need both the magnitude and argument (angle), calculated using the real and imaginary parts.

1. Resolve the following operations between complex numbers expressing the answer in rectangular form:
(4 + 2i) / (3 + 8i) - [(3 + 2i)^2 * (8 + i)]

2. Transform the following complex number from its rectangular form to its polar form: -1 - i

3. Transform the following complex number from its polar form to its rectangular form: 5√2(cos 135° + i sin 135°)

4. Perform the following multiplication between complex numbers in polar form: 4(cos 45° + i sin 45°) * 2√2(cos 180° + i sin 180°)

5. Perform the following division between complex numbers in polar form: [4(cos 160° + i sin 160°)] / [16(cos 100° + i sin 100°)]

Solve complex number problems involving division, multiplication, and transformations between rectangular and polar forms. Learn key concepts such as De Moivre's Theorem, operations with complex numbers, and the conversion between forms. Suitable for students in Grades 11-12 and undergraduates.

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