Multiply $ \dfrac{2}{5} $ by $ m^2 $ to get $ \dfrac{ 2m^2 }{ 5 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{2}{5} \cdot m^2 & \xlongequal{\text{Step 1}} \frac{2}{5} \cdot \frac{m^2}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 2 \cdot m^2 }{ 5 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 2m^2 }{ 5 } \end{aligned} $$

Multiply $ \dfrac{1}{3} $ by $ m $ to get $ \dfrac{ m }{ 3 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{1}{3} \cdot m & \xlongequal{\text{Step 1}} \frac{1}{3} \cdot \frac{m}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 1 \cdot m }{ 3 \cdot 1 } \xlongequal{\text{Step 3}} \frac{ m }{ 3 } \end{aligned} $$

Multiply $ \dfrac{1}{2} $ by $ n^2 $ to get $ \dfrac{ n^2 }{ 2 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

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

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Multiply $ \dfrac{2}{5} $ by $ m^2 $ to get $ \dfrac{ 2m^2 }{ 5 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{2}{5} \cdot m^2 & \xlongequal{\text{Step 1}} \frac{2}{5} \cdot \frac{m^2}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 2 \cdot m^2 }{ 5 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 2m^2 }{ 5 } \end{aligned} $$

Multiply $ \dfrac{m}{3} $ by $ n $ to get $ \dfrac{ mn }{ 3 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{m}{3} \cdot n & \xlongequal{\text{Step 1}} \frac{m}{3} \cdot \frac{n}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ m \cdot n }{ 3 \cdot 1 } \xlongequal{\text{Step 3}} \frac{ mn }{ 3 } \end{aligned} $$

Multiply $ \dfrac{1}{2} $ by $ n^2 $ to get $ \dfrac{ n^2 }{ 2 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Add $ \dfrac{3m^2}{2} $ and $ 2n^2 $ to get $ \dfrac{ \color{purple}{ 3m^2+4n^2 } }{ 2 }$.

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

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

Add $ \dfrac{2m^2}{5} $ and $ \dfrac{mn}{3} $ to get $ \dfrac{ \color{purple}{ 6m^2+5mn } }{ 15 }$.

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}{ 3 }$ and the second by $\color{blue}{ 5 }$.

$$ \begin{aligned} \frac{2m^2}{5} + \frac{mn}{3} & = \frac{ 2m^2 \cdot \color{blue}{ 3 }}{ 5 \cdot \color{blue}{ 3 }} + \frac{ mn \cdot \color{blue}{ 5 }}{ 3 \cdot \color{blue}{ 5 }} = \\[1ex] &=\frac{ \color{purple}{ 6m^2 } }{ 15 } + \frac{ \color{purple}{ 5mn } }{ 15 }=\frac{ \color{purple}{ 6m^2+5mn } }{ 15 } \end{aligned} $$

Multiply $ \dfrac{1}{2} $ by $ n^2 $ to get $ \dfrac{ n^2 }{ 2 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Subtract $mn$ from $ \dfrac{3m^2+4n^2}{2} $ to get $ \dfrac{ \color{purple}{ 3m^2-2mn+4n^2 } }{ 2 }$.

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

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

Subtract $ \dfrac{n^2}{2} $ from $ \dfrac{6m^2+5mn}{15} $ to get $ \dfrac{ \color{purple}{ 12m^2+10mn-15n^2 } }{ 30 }$.

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}{ 2 }$ and the second by $\color{blue}{ 15 }$.

$$ \begin{aligned} \frac{6m^2+5mn}{15} - \frac{n^2}{2} & = \frac{ \left( 6m^2+5mn \right) \cdot \color{blue}{ 2 }}{ 15 \cdot \color{blue}{ 2 }} - \frac{ n^2 \cdot \color{blue}{ 15 }}{ 2 \cdot \color{blue}{ 15 }} = \\[1ex] &=\frac{ \color{purple}{ 12m^2+10mn } }{ 30 } - \frac{ \color{purple}{ 15n^2 } }{ 30 }=\frac{ \color{purple}{ 12m^2+10mn-15n^2 } }{ 30 } \end{aligned} $$

Subtract $mn$ from $ \dfrac{3m^2+4n^2}{2} $ to get $ \dfrac{ \color{purple}{ 3m^2-2mn+4n^2 } }{ 2 }$.

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

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

Multiply $ \dfrac{12m^2+10mn-15n^2}{30} $ by $ \dfrac{3m^2-2mn+4n^2}{2} $ to get $ \dfrac{36m^4+6m^3n-17m^2n^2+70mn^3-60n^4}{60} $.

Step 1: Multiply numerators and denominators.

Step 2: Simplify numerator and denominator.

$$ \begin{aligned} \frac{12m^2+10mn-15n^2}{30} \cdot \frac{3m^2-2mn+4n^2}{2} & \xlongequal{\text{Step 1}} \frac{ \left( 12m^2+10mn-15n^2 \right) \cdot \left( 3m^2-2mn+4n^2 \right) }{ 30 \cdot 2 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ 36m^4-24m^3n+48m^2n^2+30m^3n-20m^2n^2+40mn^3-45m^2n^2+30mn^3-60n^4 }{ 60 } = \frac{36m^4+6m^3n-17m^2n^2+70mn^3-60n^4}{60} \end{aligned} $$