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 $-(2a^2 + b - 3b^2) = -(2a^2+ 3b - b)$