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The problem requires finding the domain and simplifying the following expression:

\[
\frac{b}{2a^2 - ab} - \frac{4a}{2ab - b^2}
\]

### Step 1: Domain of the Expression

The domain is determined by any restrictions that make the denominator zero. We'll analyze the denominators of each fraction to find where they are equal to zero.

1. For the first term, the denominator is \( 2a^2 - ab \). Set this equal to zero to find restrictions:
   \[
   2a^2 - ab = 0 \quad \Rightarrow \quad a(2a - b) = 0
   \]
   This means \( a = 0 \) or \( 2a = b \) (or equivalently \( a = \frac{b}{2} \)).

2. For the second term, the denominator is \( 2ab - b^2 \). Set this equal to zero:
   \[
   2ab - b^2 = 0 \quad \Rightarrow \quad b(2a - b) = 0
   \]
   This gives \( b = 0 \) or \( 2a = b \) (or \( a = \frac{b}{2} \)).

Therefore, the expression is undefined when:
- \( a = 0 \)
- \( b = 0 \)
- \( a = \frac{b}{2} \)

Thus, the domain of the expression is all real numbers except \( a = 0 \), \( b = 0 \), and \( a = \frac{b}{2} \).

### Step 2: Simplifying the Expression

We now simplify:

\[
\frac{b}{2a^2 - ab} - \frac{4a}{2ab - b^2}
\]

Since the denominators are similar, we factor both denominators:
\[
2a^2 - ab = a(2a - b), \quad 2ab - b^2 = b(2a - b)
\]

Thus, the expression becomes:
\[
\frac{b}{a(2a - b)} - \frac{4a}{b(2a - b)}
\]

Now, combine these terms under a common denominator:
\[
\frac{b}{a(2a - b)} - \frac{4a}{b(2a - b)} = \frac{b \cdot b - 4a \cdot a}{ab(2a - b)} = \frac{b^2 - 4a^2}{ab(2a - b)}
\]

The numerator is a difference of squares:
\[
b^2 - 4a^2 = (b - 2a)(b + 2a)
\]

Thus, the simplified expression is:
\[
\frac{(b - 2a)(b + 2a)}{ab(2a - b)}
\]

Notice that \( (2a - b) = -(b - 2a) \), so the final simplified form is:
\[
\frac{-(b + 2a)}{ab}
\]

### Final Answer:
The simplified expression is:
\[
\frac{-(b + 2a)}{ab}
\]
And the domain is all real numbers except \( a = 0 \), \( b = 0 \), and \( a = \frac{b}{2} \).

Would you like further clarification on any part of the solution or a detailed breakdown? Here are some related questions:

1. How do you find the domain of rational expressions?
2. What are the steps involved in simplifying complex rational expressions?
3. Why does the factorization of the difference of squares simplify the expression?
4. How do common denominators help in combining rational terms?
5. What happens if we don't simplify the expression properly?

**Tip**: Always check for common factors in both the numerator and denominator before simplifying complex rational expressions.

Find the domain and simplify the expression: \( \frac{b}{2a^2 - ab} - \frac{4a}{2ab - b^2} \)

Learn how to find the domain and simplify the rational expression \( \frac{b}{2a^2 - ab} - \frac{4a}{2ab - b^2} \). This problem involves key algebraic concepts such as factorization and rational expressions. Suitable for high school students in Grades 10-12.

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