Multiply $ \color{blue}{2b^2} $ by $ \left( a^2-a\right) $ $$ \color{blue}{2b^2} \cdot \left( a^2-a\right) = 2a^2b^2-2ab^2 $$

Multiply $ab^2$ by $ \dfrac{a^2+a-2}{2a^2b^2-2ab^2} $ to get $ \dfrac{ a^3b^2+a^2b^2-2ab^2 }{ 2a^2b^2-2ab^2 } $.

Step 1: Write $ ab^2 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} ab^2 \cdot \frac{a^2+a-2}{2a^2b^2-2ab^2} & \xlongequal{\text{Step 1}} \frac{ab^2}{\color{red}{1}} \cdot \frac{a^2+a-2}{2a^2b^2-2ab^2} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ ab^2 \cdot \left( a^2+a-2 \right) }{ 1 \cdot \left( 2a^2b^2-2ab^2 \right) } \xlongequal{\text{Step 3}} \frac{ a^3b^2+a^2b^2-2ab^2 }{ 2a^2b^2-2ab^2 } \end{aligned} $$

Simplify $ \dfrac{a^2+a-2}{2a-2} $ to $ \dfrac{a+2}{2} $.

Factor both the denominator and the numerator, then cancel the common factor. $\color{blue}{a-1}$.

$$ \begin{aligned} \frac{a^2+a-2}{2a-2} & =\frac{ \left( a+2 \right) \cdot \color{blue}{ \left( a-1 \right) }}{ 2 \cdot \color{blue}{ \left( a-1 \right) }} = \\[1ex] &= \frac{a+2}{2} \end{aligned} $$