To rationalize the denominator of the expression:

$\frac{2 - 3\sqrt{2}}{1 - \sqrt{2}}$

we multiply both the numerator and denominator by the conjugate of the denominator. The conjugate of $1 - \sqrt{2}$ is $1 + \sqrt{2}$.

Step 1: Multiply by the conjugate

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

Step 2: Simplify the denominator using the difference of squares formula

$(1 - \sqrt{2})(1 + \sqrt{2}) = 1^2 - (\sqrt{2})^2 = 1 - 2 = -1$

Step 3: Expand the numerator

Now we expand $(2 - 3\sqrt{2})(1 + \sqrt{2})$:

$2(1 + \sqrt{2}) - 3\sqrt{2}(1 + \sqrt{2}) = 2 + 2\sqrt{2} - 3\sqrt{2} - 3(\sqrt{2})^2$

Since $(\sqrt{2})^2 = 2$, we get:

$2 + 2\sqrt{2} - 3\sqrt{2} - 3(2) = 2 + (2\sqrt{2} - 3\sqrt{2}) - 6$

Simplify the like terms:

$2 + (-1\sqrt{2}) - 6 = -4 - \sqrt{2}$

Step 4: Combine the numerator and denominator

Now the expression becomes:

$\frac{-4 - \sqrt{2}}{-1}$

Step 5: Simplify the fraction

Dividing both terms in the numerator by $-1$:

$4 + \sqrt{2}$

Thus, the rationalized expression is:

$\boxed{4 + \sqrt{2}}$

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Would you like more details on any of the steps? Here are some related questions:

1. How do you determine the conjugate of a binomial involving square roots?
2. What is the difference of squares, and how does it simplify expressions?
3. Could you explain more examples of rationalizing denominators?
4. What happens when you have a different radical, like $\sqrt{3}$, in the denominator?
5. How does rationalizing denominators help in simplifying complex expressions?

Tip: When rationalizing a denominator, always multiply by the conjugate of the denominator if it contains square roots.