To solve the expression \(\frac{3 - 2i}{-1 + i}\), we will multiply both the numerator and the denominator by the conjugate of the denominator to simplify it.

### Step 1: Identify the conjugate
The conjugate of \(-1 + i\) is \(-1 - i\).

### Step 2: Multiply the numerator and denominator by the conjugate
\[
\frac{3 - 2i}{-1 + i} \times \frac{-1 - i}{-1 - i} = \frac{(3 - 2i)(-1 - i)}{(-1 + i)(-1 - i)}
\]

### Step 3: Expand the numerator and denominator
Expand the numerator:
\[
(3 - 2i)(-1 - i) = 3(-1) + 3(-i) - 2i(-1) - 2i(i) = -3 - 3i + 2i - 2i^2
\]

Simplify the expression using \(i^2 = -1\):
\[
-3 - 3i + 2i - 2(-1) = -3 - i + 2 = -1 - i
\]

Expand the denominator:
\[
(-1 + i)(-1 - i) = (-1)^2 - (i)^2 = 1 - (-1) = 1 + 1 = 2
\]

### Step 4: Simplify the expression
\[
\frac{-1 - i}{2} = -\frac{1}{2} - \frac{1}{2}i
\]

So, the simplified expression is \(\boxed{-\frac{1}{2} - \frac{1}{2}i}\).

Would you like details on any step? Here are some related questions you might find useful:

1. How do you find the conjugate of a complex number?
2. What is the significance of multiplying by the conjugate in complex division?
3. How can you expand expressions involving complex numbers?
4. Why does multiplying a complex number by its conjugate result in a real number?
5. What are some other examples of simplifying complex number fractions?

**Tip:** When dividing complex numbers, always multiply by the conjugate to eliminate the imaginary part from the denominator.

Learn how to simplify the complex division problem (3-2i)/(-1+i) step-by-step. This problem involves complex numbers and the process of multiplying by the conjugate to eliminate the imaginary part from the denominator. Suitable for advanced high school students and anyone studying complex number division.

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