Multiply $ \dfrac{25}{2} $ by $ m^3 $ to get $ \dfrac{ 25m^3 }{ 2 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{25}{2} \cdot m^3 & \xlongequal{\text{Step 1}} \frac{25}{2} \cdot \frac{m^3}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 25 \cdot m^3 }{ 2 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 25m^3 }{ 2 } \end{aligned} $$

Add $m^2-10m$ and $ \dfrac{25m^3}{2} $ to get $ \dfrac{ \color{purple}{ 25m^3+2m^2-20m } }{ 2 }$.

Step 1: Write $ m^2-10m $ 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 $\color{blue}{ 2 }$.

$$ \begin{aligned} m^2-10m+ \frac{25m^3}{2} & \xlongequal{\text{Step 1}} \frac{m^2-10m}{\color{red}{1}} + \frac{25m^3}{2} = \frac{ \left( m^2-10m \right) \cdot \color{blue}{ 2 }}{ 1 \cdot \color{blue}{ 2 }} + \frac{ 25m^3 }{ 2 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 2m^2-20m } }{ 2 } + \frac{ \color{purple}{ 25m^3 } }{ 2 }=\frac{ \color{purple}{ 25m^3+2m^2-20m } }{ 2 } \end{aligned} $$

Subtract $15m^2$ from $ \dfrac{25m^3+2m^2-20m}{2} $ to get $ \dfrac{ \color{purple}{ 25m^3-28m^2-20m } }{ 2 }$.

Step 1: Write $ 15m^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 $\color{blue}{2}$.

$$ \begin{aligned} \frac{25m^3+2m^2-20m}{2} -15m^2 & \xlongequal{\text{Step 1}} \frac{25m^3+2m^2-20m}{2} - \frac{15m^2}{\color{red}{1}} = \frac{ 25m^3+2m^2-20m }{ 2 } - \frac{ 15m^2 \cdot \color{blue}{ 2 }}{ 1 \cdot \color{blue}{ 2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 25m^3+2m^2-20m } }{ 2 } - \frac{ \color{purple}{ 30m^2 } }{ 2 }=\frac{ \color{purple}{ 25m^3-28m^2-20m } }{ 2 } \end{aligned} $$

Add $ \dfrac{25m^3-28m^2-20m}{2} $ and $ 25m $ to get $ \dfrac{ \color{purple}{ 25m^3-28m^2+30m } }{ 2 }$.

Step 1: Write $ 25m $ 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 second fraction by $\color{blue}{2}$.

$$ \begin{aligned} \frac{25m^3-28m^2-20m}{2} +25m & \xlongequal{\text{Step 1}} \frac{25m^3-28m^2-20m}{2} + \frac{25m}{\color{red}{1}} = \frac{ 25m^3-28m^2-20m }{ 2 } + \frac{ 25m \cdot \color{blue}{ 2 }}{ 1 \cdot \color{blue}{ 2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 25m^3-28m^2-20m } }{ 2 } + \frac{ \color{purple}{ 50m } }{ 2 }=\frac{ \color{purple}{ 25m^3-28m^2+30m } }{ 2 } \end{aligned} $$