Multiply each term of $ \left( \color{blue}{k+1}\right) $ by each term in $ \left( k+2\right) $.

$$ \left( \color{blue}{k+1}\right) \cdot \left( k+2\right) = k^2+2k+k+2 $$

Combine like terms:

$$ k^2+ \color{blue}{2k} + \color{blue}{k} +2 = k^2+ \color{blue}{3k} +2 $$

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

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Multiply $ \dfrac{k}{3} $ by $ k+1 $ to get $ \dfrac{ k^2+k }{ 3 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

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

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Add $ \dfrac{k^3+3k^2+2k}{3} $ and $ k^2+3k+2 $ to get $ \dfrac{ \color{purple}{ k^3+6k^2+11k+6 } }{ 3 }$.

Step 1: Write $ k^2+3k+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 $\color{blue}{3}$.

$$ \begin{aligned} \frac{k^3+3k^2+2k}{3} +k^2+3k+2 & \xlongequal{\text{Step 1}} \frac{k^3+3k^2+2k}{3} + \frac{k^2+3k+2}{\color{red}{1}} = \frac{ k^3+3k^2+2k }{ 3 } + \frac{ \left( k^2+3k+2 \right) \cdot \color{blue}{ 3 }}{ 1 \cdot \color{blue}{ 3 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ k^3+3k^2+2k } }{ 3 } + \frac{ \color{purple}{ 3k^2+9k+6 } }{ 3 }=\frac{ \color{purple}{ k^3+6k^2+11k+6 } }{ 3 } \end{aligned} $$