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

To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $\color{blue}{ 3 }$.

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

Add $ \dfrac{x^3+3x^2}{6} $ and $ \dfrac{x^4}{24} $ to get $ \dfrac{ \color{purple}{ x^4+4x^3+12x^2 } }{ 24 }$.

To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $\color{blue}{ 4 }$.

$$ \begin{aligned} \frac{x^3+3x^2}{6} + \frac{x^4}{24} & = \frac{ \left( x^3+3x^2 \right) \cdot \color{blue}{ 4 }}{ 6 \cdot \color{blue}{ 4 }} + \frac{ x^4 }{ 24 } = \\[1ex] &=\frac{ \color{purple}{ 4x^3+12x^2 } }{ 24 } + \frac{ \color{purple}{ x^4 } }{ 24 }=\frac{ \color{purple}{ x^4+4x^3+12x^2 } }{ 24 } \end{aligned} $$