Multiply $6$ by $ \dfrac{b}{3+b} $ to get $ \dfrac{6b}{b+3} $.

Step 1: Write $ 6 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 6 \cdot \frac{b}{3+b} & \xlongequal{\text{Step 1}} \frac{6}{\color{red}{1}} \cdot \frac{b}{3+b} \xlongequal{\text{Step 2}} \frac{ 6 \cdot b }{ 1 \cdot \left( 3+b \right) } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 6b }{ 3+b } = \frac{6b}{b+3} \end{aligned} $$

Subtract $ \dfrac{6b}{3+b} $ from $ 6 $ to get $ \dfrac{ \color{purple}{ 18 } }{ b+3 }$.

Step 1: Write $ 6 $ 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 $\color{blue}{ b+3 }$.

$$ \begin{aligned} 6- \frac{6b}{3+b} & \xlongequal{\text{Step 1}} \frac{6}{\color{red}{1}} - \frac{6b}{3+b} = \frac{ 6 \cdot \color{blue}{ \left( b+3 \right) }}{ 1 \cdot \color{blue}{ \left( b+3 \right) }} - \frac{ 6b }{ 3+b } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 6b+18 } }{ b+3 } - \frac{ \color{purple}{ 6b } }{ b+3 }=\frac{ \color{purple}{ 18 } }{ b+3 } \end{aligned} $$

Multiply $3$ by $ \dfrac{6b}{b+3} $ to get $ \dfrac{ 18b }{ b+3 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 3 \cdot \frac{6b}{b+3} & \xlongequal{\text{Step 1}} \frac{3}{\color{red}{1}} \cdot \frac{6b}{b+3} \xlongequal{\text{Step 2}} \frac{ 3 \cdot 6b }{ 1 \cdot \left( b+3 \right) } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 18b }{ b+3 } \end{aligned} $$

Subtract $ \dfrac{6b}{3+b} $ from $ 6 $ to get $ \dfrac{ \color{purple}{ 18 } }{ b+3 }$.

Step 1: Write $ 6 $ 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 $\color{blue}{ b+3 }$.

$$ \begin{aligned} 6- \frac{6b}{3+b} & \xlongequal{\text{Step 1}} \frac{6}{\color{red}{1}} - \frac{6b}{3+b} = \frac{ 6 \cdot \color{blue}{ \left( b+3 \right) }}{ 1 \cdot \color{blue}{ \left( b+3 \right) }} - \frac{ 6b }{ 3+b } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 6b+18 } }{ b+3 } - \frac{ \color{purple}{ 6b } }{ b+3 }=\frac{ \color{purple}{ 18 } }{ b+3 } \end{aligned} $$

Divide $ \dfrac{18b}{b+3} $ by $ \dfrac{18}{b+3} $ to get $ \dfrac{ 18b }{ 18 } $.

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

Step 2: Cancel $ \color{red}{ b+3 } $ in first and second fraction.

Step 3: Multiply numerators and denominators.

Step 4: Simplify numerator and denominator.

$$ \begin{aligned} \frac{ \frac{18b}{b+3} }{ \frac{\color{blue}{18}}{\color{blue}{b+3}} } & \xlongequal{\text{Step 1}} \frac{18b}{b+3} \cdot \frac{\color{blue}{b+3}}{\color{blue}{18}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{18b}{\color{red}{1}} \cdot \frac{\color{red}{1}}{18} \xlongequal{\text{Step 3}} \frac{ 18b \cdot 1 }{ 1 \cdot 18 } \xlongequal{\text{Step 4}} \frac{ 18b }{ 18 } \end{aligned} $$