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

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Subtract $ \dfrac{5x^2}{2} $ from $ 2x^2+9x $ to get $ \dfrac{ \color{purple}{ -x^2+18x } }{ 2 }$.

Step 1: Write $ 2x^2+9x $ 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 first fraction by $\color{blue}{ 2 }$.

$$ \begin{aligned} 2x^2+9x- \frac{5x^2}{2} & \xlongequal{\text{Step 1}} \frac{2x^2+9x}{\color{red}{1}} - \frac{5x^2}{2} = \frac{ \left( 2x^2+9x \right) \cdot \color{blue}{ 2 }}{ 1 \cdot \color{blue}{ 2 }} - \frac{ 5x^2 }{ 2 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 4x^2+18x } }{ 2 } - \frac{ \color{purple}{ 5x^2 } }{ 2 }=\frac{ \color{purple}{ -x^2+18x } }{ 2 } \end{aligned} $$

Add $ \dfrac{-x^2+18x}{2} $ and $ 16x $ to get $ \dfrac{ \color{purple}{ -x^2+50x } }{ 2 }$.

Step 1: Write $ 16x $ 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 $\color{blue}{2}$.

$$ \begin{aligned} \frac{-x^2+18x}{2} +16x & \xlongequal{\text{Step 1}} \frac{-x^2+18x}{2} + \frac{16x}{\color{red}{1}} = \frac{ -x^2+18x }{ 2 } + \frac{ 16x \cdot \color{blue}{ 2 }}{ 1 \cdot \color{blue}{ 2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ -x^2+18x } }{ 2 } + \frac{ \color{purple}{ 32x } }{ 2 }=\frac{ \color{purple}{ -x^2+50x } }{ 2 } \end{aligned} $$

Add $ \dfrac{-x^2+50x}{2} $ and $ 30 $ to get $ \dfrac{ \color{purple}{ -x^2+50x+60 } }{ 2 }$.

Step 1: Write $ 30 $ 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 $\color{blue}{2}$.

$$ \begin{aligned} \frac{-x^2+50x}{2} +30 & \xlongequal{\text{Step 1}} \frac{-x^2+50x}{2} + \frac{30}{\color{red}{1}} = \frac{ -x^2+50x }{ 2 } + \frac{ 30 \cdot \color{blue}{ 2 }}{ 1 \cdot \color{blue}{ 2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ -x^2+50x } }{ 2 } + \frac{ \color{purple}{ 60 } }{ 2 }=\frac{ \color{purple}{ -x^2+50x+60 } }{ 2 } \end{aligned} $$