To divide the complex number $\frac{3 + 2i}{1 - i}$, we use the method of multiplying both the numerator and the denominator by the conjugate of the denominator.

The conjugate of $1 - i$ is $1 + i$. Let's proceed step by step:

Step 1: Multiply numerator and denominator by the conjugate of the denominator.

$\frac{3 + 2i}{1 - i} \times \frac{1 + i}{1 + i} = \frac{(3 + 2i)(1 + i)}{(1 - i)(1 + i)}$

Step 2: Simplify the denominator.

$(1 - i)(1 + i) = 1^2 - i^2 = 1 - (-1) = 2$

Step 3: Multiply the numerators.

$(3 + 2i)(1 + i) = 3(1) + 3(i) + 2i(1) + 2i(i)$ $= 3 + 3i + 2i + 2i^2 = 3 + 5i + 2(-1) = 3 + 5i - 2 = 1 + 5i$

Step 4: Combine the results.

$\frac{1 + 5i}{2}$

Step 5: Write the result in standard form.

$= \frac{1}{2} + \frac{5}{2}i$

Thus, the result of dividing $3 + 2i$ by $1 - i$ is: $\frac{1}{2} + \frac{5}{2}i$

Would you like more details or explanations?

Here are 5 related questions:

1. How do you multiply complex numbers in polar form?
2. What is the significance of the conjugate in complex number division?
3. How do you convert a complex number from Cartesian to polar form?
4. Can you find the magnitude and argument of the complex number $3 + 2i$?
5. How do you solve equations involving complex numbers?

Tip: Always simplify complex number operations by using conjugates to eliminate imaginary parts from denominators.