Multiply $ \dfrac{1}{r} $ by $ h^2+a^2 $ to get $ \dfrac{a^2+h^2}{r} $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Add $ \dfrac{-a^2-h^2}{r} $ and $ \dfrac{1}{h} $ to get $ \dfrac{ \color{purple}{ -a^2h-h^3+r } }{ hr }$.

To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $ \color{blue}{ h }$ and the second by $\color{blue}{ r }$.

$$ \begin{aligned} \frac{-a^2-h^2}{r} + \frac{1}{h} & = \frac{ \left( -a^2-h^2 \right) \cdot \color{blue}{ h }}{ r \cdot \color{blue}{ h }} + \frac{ 1 \cdot \color{blue}{ r }}{ h \cdot \color{blue}{ r }} = \\[1ex] &=\frac{ \color{purple}{ -a^2h-h^3 } }{ hr } + \frac{ \color{purple}{ r } }{ hr }=\frac{ \color{purple}{ -a^2h-h^3+r } }{ hr } \end{aligned} $$