Multiply $ \dfrac{5}{6} $ by $ x^2 $ to get $ \dfrac{ 5x^2 }{ 6 } $.

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.

$$ \begin{aligned} \frac{5}{6} \cdot x^2 & \xlongequal{\text{Step 1}} \frac{5}{6} \cdot \frac{x^2}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 5 \cdot x^2 }{ 6 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 5x^2 }{ 6 } \end{aligned} $$

Subtract $ \dfrac{5x^2}{6} $ from $ 6x^2+13x $ to get $ \dfrac{ \color{purple}{ 31x^2+78x } }{ 6 }$.

Step 1: Write $ 6x^2+13x $ 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 $\color{blue}{ 6 }$.

$$ \begin{aligned} 6x^2+13x- \frac{5x^2}{6} & \xlongequal{\text{Step 1}} \frac{6x^2+13x}{\color{red}{1}} - \frac{5x^2}{6} = \frac{ \left( 6x^2+13x \right) \cdot \color{blue}{ 6 }}{ 1 \cdot \color{blue}{ 6 }} - \frac{ 5x^2 }{ 6 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 36x^2+78x } }{ 6 } - \frac{ \color{purple}{ 5x^2 } }{ 6 }=\frac{ \color{purple}{ 31x^2+78x } }{ 6 } \end{aligned} $$

Subtract $23x$ from $ \dfrac{31x^2+78x}{6} $ to get $ \dfrac{ \color{purple}{ 31x^2-60x } }{ 6 }$.

Step 1: Write $ 23x $ 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 $\color{blue}{6}$.

$$ \begin{aligned} \frac{31x^2+78x}{6} -23x & \xlongequal{\text{Step 1}} \frac{31x^2+78x}{6} - \frac{23x}{\color{red}{1}} = \frac{ 31x^2+78x }{ 6 } - \frac{ 23x \cdot \color{blue}{ 6 }}{ 1 \cdot \color{blue}{ 6 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 31x^2+78x } }{ 6 } - \frac{ \color{purple}{ 138x } }{ 6 }=\frac{ \color{purple}{ 31x^2-60x } }{ 6 } \end{aligned} $$

Add $ \dfrac{31x^2-60x}{6} $ and $ 7 $ to get $ \dfrac{ \color{purple}{ 31x^2-60x+42 } }{ 6 }$.

Step 1: Write $ 7 $ 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 $\color{blue}{6}$.

$$ \begin{aligned} \frac{31x^2-60x}{6} +7 & \xlongequal{\text{Step 1}} \frac{31x^2-60x}{6} + \frac{7}{\color{red}{1}} = \frac{ 31x^2-60x }{ 6 } + \frac{ 7 \cdot \color{blue}{ 6 }}{ 1 \cdot \color{blue}{ 6 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 31x^2-60x } }{ 6 } + \frac{ \color{purple}{ 42 } }{ 6 }=\frac{ \color{purple}{ 31x^2-60x+42 } }{ 6 } \end{aligned} $$