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To convert the given polar equation \( r = 4 \sin \theta - 6 \cos \theta \) into rectangular form, we can use the following standard polar-to-rectangular conversions:

- \( r = \sqrt{x^2 + y^2} \)
- \( x = r \cos \theta \)
- \( y = r \sin \theta \)

We will also need the identity relations for the trigonometric functions:

- \( \sin \theta = \frac{y}{r} \)
- \( \cos \theta = \frac{x}{r} \)

### Step-by-Step Conversion:

1. Start with the given equation:
   \[
   r = 4 \sin \theta - 6 \cos \theta
   \]
   
2. Replace \( \sin \theta \) and \( \cos \theta \) using their rectangular equivalents:
   \[
   r = 4 \left(\frac{y}{r}\right) - 6 \left(\frac{x}{r}\right)
   \]

3. Multiply both sides by \( r \) to eliminate the denominator:
   \[
   r^2 = 4y - 6x
   \]
   
4. Now, recall that \( r^2 = x^2 + y^2 \), so we substitute this into the equation:
   \[
   x^2 + y^2 = 4y - 6x
   \]

5. Rearrange the equation to set everything to one side:
   \[
   x^2 + y^2 + 6x - 4y = 0
   \]

6. Complete the square for both the \( x \)-terms and \( y \)-terms:
   
   - For \( x \), take half of 6 and square it: \( \left(\frac{6}{2}\right)^2 = 9 \), so we add and subtract 9.
   - For \( y \), take half of 4 and square it: \( \left(\frac{4}{2}\right)^2 = 4 \), so we add and subtract 4.
   
   The equation becomes:
   \[
   (x^2 + 6x + 9) + (y^2 - 4y + 4) = 9 + 4
   \]
   Simplifying:
   \[
   (x + 3)^2 + (y - 2)^2 = 13
   \]

This is the equation of a circle with center \( (-3, 2) \) and radius \( \sqrt{13} \).

### Conclusion:
The correct rectangular form is:
\[
(x + 3)^2 + (y - 2)^2 = 13
\]

Thus, the correct answer is **C**.

Would you like further details or have any questions?

Here are 5 related questions for deeper understanding:
1. How do you convert between polar and rectangular coordinates?
2. What is the geometric interpretation of the equation \( r = 4 \sin \theta - 6 \cos \theta \)?
3. How does completing the square help in solving conic sections?
4. Can you generalize how to find the center and radius of a circle from its equation?
5. What other types of equations can be converted between polar and rectangular forms?

**Tip:** Remember that completing the square is a useful technique when transforming quadratic equations into standard forms like circles or parabolas.

Write the polar equation r = 4sinθ - 6cosθ in rectangular form.

Learn how to convert the polar equation r = 4sinθ - 6cosθ to rectangular form, involving polar-to-rectangular transformations and completing the square. Suitable for advanced high school or college-level students.

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