Multiply $10$ by $ \dfrac{p}{p-2} $ to get $ \dfrac{ 10p }{ p-2 } $.

Step 1: Write $ 10 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 10 \cdot \frac{p}{p-2} & \xlongequal{\text{Step 1}} \frac{10}{\color{red}{1}} \cdot \frac{p}{p-2} \xlongequal{\text{Step 2}} \frac{ 10 \cdot p }{ 1 \cdot \left( p-2 \right) } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 10p }{ p-2 } \end{aligned} $$

Add $p-5$ and $ \dfrac{10p}{p-2} $ to get $ \dfrac{ \color{purple}{ p^2+3p+10 } }{ p-2 }$.

Step 1: Write $ p-5 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

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

$$ \begin{aligned} p-5+ \frac{10p}{p-2} & \xlongequal{\text{Step 1}} \frac{p-5}{\color{red}{1}} + \frac{10p}{p-2} = \frac{ \left( p-5 \right) \cdot \color{blue}{ \left( p-2 \right) }}{ 1 \cdot \color{blue}{ \left( p-2 \right) }} + \frac{ 10p }{ p-2 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ p^2-2p-5p+10 } }{ p-2 } + \frac{ \color{purple}{ 10p } }{ p-2 }=\frac{ \color{purple}{ p^2+3p+10 } }{ p-2 } \end{aligned} $$