Add $ \dfrac{1}{4} $ and $ x $ to get $ \dfrac{ \color{purple}{ 4x+1 } }{ 4 }$.

Step 1: Write $ x $ 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 $\color{blue}{4}$.

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

Add $ \dfrac{4x+1}{4} $ and $ \dfrac{3}{5} $ to get $ \dfrac{ \color{purple}{ 20x+17 } }{ 20 }$.

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}{ 5 }$ and the second by $\color{blue}{ 4 }$.

$$ \begin{aligned} \frac{4x+1}{4} + \frac{3}{5} & = \frac{ \left( 4x+1 \right) \cdot \color{blue}{ 5 }}{ 4 \cdot \color{blue}{ 5 }} + \frac{ 3 \cdot \color{blue}{ 4 }}{ 5 \cdot \color{blue}{ 4 }} = \\[1ex] &=\frac{ \color{purple}{ 20x+5 } }{ 20 } + \frac{ \color{purple}{ 12 } }{ 20 }=\frac{ \color{purple}{ 20x+17 } }{ 20 } \end{aligned} $$