Add $c$ and $ \dfrac{125}{c^2} $ to get $ \dfrac{ \color{purple}{ c^3+125 } }{ c^2 }$.

Step 1: Write $ c $ 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 $\color{blue}{ c^2 }$.

$$ \begin{aligned} c+ \frac{125}{c^2} & \xlongequal{\text{Step 1}} \frac{c}{\color{red}{1}} + \frac{125}{c^2} = \frac{ c \cdot \color{blue}{ c^2 }}{ 1 \cdot \color{blue}{ c^2 }} + \frac{ 125 }{ c^2 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ c^3 } }{ c^2 } + \frac{ \color{purple}{ 125 } }{ c^2 }=\frac{ \color{purple}{ c^3+125 } }{ c^2 } \end{aligned} $$

Add $1$ and $ \dfrac{5}{c} $ to get $ \dfrac{ \color{purple}{ c+5 } }{ c }$.

Step 1: Write $ 1 $ 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 $\color{blue}{ c }$.

$$ \begin{aligned} 1+ \frac{5}{c} & \xlongequal{\text{Step 1}} \frac{1}{\color{red}{1}} + \frac{5}{c} = \frac{ 1 \cdot \color{blue}{ c }}{ 1 \cdot \color{blue}{ c }} + \frac{ 5 }{ c } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ c } }{ c } + \frac{ \color{purple}{ 5 } }{ c }=\frac{ \color{purple}{ c+5 } }{ c } \end{aligned} $$

Divide $ \dfrac{c^3+125}{c^2} $ by $ \dfrac{c+5}{c} $ to get $ \dfrac{ c^3-5c^2+25c }{ c^2 } $.

Step 1: To divide rational expressions, multiply the first fraction by the reciprocal of the second fraction.

Step 2: Factor numerators and denominators.

Step 3: Cancel common factors.

Step 4: Multiply numerators and denominators.

Step 5: Simplify numerator and denominator.

$$ \begin{aligned} \frac{ \frac{c^3+125}{c^2} }{ \frac{\color{blue}{c+5}}{\color{blue}{c}} } & \xlongequal{\text{Step 1}} \frac{c^3+125}{c^2} \cdot \frac{\color{blue}{c}}{\color{blue}{c+5}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( c^2-5c+25 \right) \cdot \color{blue}{ \left( c+5 \right) } }{ c^2 } \cdot \frac{ c }{ 1 \cdot \color{blue}{ \left( c+5 \right) } } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ c^2-5c+25 }{ c^2 } \cdot \frac{ c }{ 1 } \xlongequal{\text{Step 4}} \frac{ \left( c^2-5c+25 \right) \cdot c }{ c^2 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 5}} \frac{ c^3-5c^2+25c }{ c^2 } \end{aligned} $$