Subtract $ \dfrac{a}{a} $ from $ a^3-a^2 $ to get $ \dfrac{ \color{purple}{ a^4-a^3-a } }{ a }$.

Step 1: Write $ a^3-a^2 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: To subtract raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $\color{blue}{ a }$.

$$ \begin{aligned} a^3-a^2- \frac{a}{a} & \xlongequal{\text{Step 1}} \frac{a^3-a^2}{\color{red}{1}} - \frac{a}{a} = \frac{ \left( a^3-a^2 \right) \cdot \color{blue}{ a }}{ 1 \cdot \color{blue}{ a }} - \frac{ a }{ a } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ a^4-a^3 } }{ a } - \frac{ \color{purple}{ a } }{ a }=\frac{ \color{purple}{ a^4-a^3-a } }{ a } \end{aligned} $$

Subtract $2$ from $ \dfrac{a^4-a^3-a}{a} $ to get $ \dfrac{ \color{purple}{ a^4-a^3-3a } }{ a }$.

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

Step 2: To subtract raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{a}$.

$$ \begin{aligned} \frac{a^4-a^3-a}{a} -2 & \xlongequal{\text{Step 1}} \frac{a^4-a^3-a}{a} - \frac{2}{\color{red}{1}} = \frac{ a^4-a^3-a }{ a } - \frac{ 2 \cdot \color{blue}{ a }}{ 1 \cdot \color{blue}{ a }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ a^4-a^3-a } }{ a } - \frac{ \color{purple}{ 2a } }{ a }=\frac{ \color{purple}{ a^4-a^3-3a } }{ a } \end{aligned} $$