Simplify the complex rational expression by the method of your choice.
StartStartFraction StartFraction 3 Over a squared EndFraction minus StartFraction 1 Over ab EndFraction minus StartFraction 2 Over b squared EndFraction OverOver StartFraction 2 Over a squared EndFraction minus StartFraction 5 Over ab EndFraction plus StartFraction 3 Over b squared EndFraction EndEndFraction
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Part 1
StartStartFraction StartFraction 3 Over a squared EndFraction minus StartFraction 1 Over ab EndFraction minus StartFraction 2 Over b squared EndFraction OverOver StartFraction 2 Over a squared EndFraction minus StartFraction 5 Over ab EndFraction plus StartFraction 3 Over b squared EndFraction EndEndFraction
equals
  
negative StartFraction 2 a plus 3 b Over 3 a minus 2 b EndFraction
 
​(Simplify your answer. Type your answer in factored​ form.)

To simplify the given complex rational expression:

\[
\frac{\frac{3}{a^2} - \frac{1}{ab} - \frac{2}{b^2}}{\frac{2}{a^2} - \frac{5}{ab} + \frac{3}{b^2}},
\]

we will first combine the fractions in the numerator and denominator by finding a common denominator for each.

### Step 1: Simplify the Numerator
The common denominator for the terms in the numerator is \( a^2 b^2 \):

\[
\frac{3}{a^2} = \frac{3b^2}{a^2 b^2}, \quad \frac{1}{ab} = \frac{b}{a^2 b^2}, \quad \frac{2}{b^2} = \frac{2a^2}{a^2 b^2}.
\]

Combining these, the numerator becomes:

\[
\frac{3b^2 - b - 2a^2}{a^2 b^2}.
\]

### Step 2: Simplify the Denominator
Similarly, the common denominator for the terms in the denominator is also \( a^2 b^2 \):

\[
\frac{2}{a^2} = \frac{2b^2}{a^2 b^2}, \quad \frac{5}{ab} = \frac{5b}{a^2 b^2}, \quad \frac{3}{b^2} = \frac{3a^2}{a^2 b^2}.
\]

Combining these, the denominator becomes:

\[
\frac{2b^2 - 5b + 3a^2}{a^2 b^2}.
\]

### Step 3: Form the Complex Fraction
The original expression simplifies to:

\[
\frac{\frac{3b^2 - b - 2a^2}{a^2 b^2}}{\frac{2b^2 - 5b + 3a^2}{a^2 b^2}} = \frac{3b^2 - b - 2a^2}{2b^2 - 5b + 3a^2}.
\]

### Step 4: Factor the Numerator and Denominator
Now, we factor both the numerator and the denominator:

- **Numerator:** \( 3b^2 - b - 2a^2 \) factors as \( -(2a^2 + b - 3b^2) = -(2a^2+ 3b - b)\)
  


Simplify the complex rational expression by the method of your choice. \( \frac{\frac{3}{a^2} - \frac{1}{ab} - \frac{2}{b^2}}{\frac{2}{a^2} - \frac{5}{ab} + \frac{3}{b^2}} = -\frac{2a + 3b}{3a - 2b} \) (Simplify your answer. Type your answer in factored form.)

Discover step-by-step methods for simplifying complex rational expressions by factoring. This problem guides you through combining fractions, simplifying complex expressions, and factoring. Ideal for students in Grades 10-12 studying algebra and rational expressions.

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