Subtract $ \dfrac{1}{4} $ from $ a $ to get $ \dfrac{ \color{purple}{ 4a-1 } }{ 4 }$.

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

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

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

Step 1: Write $ a $ 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 first fraction by $\color{blue}{ 4 }$.

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

Multiply $ \dfrac{4a-1}{4} $ by $ \dfrac{4a+1}{4} $ to get $ \dfrac{16a^2-1}{16} $.

Step 1: Multiply numerators and denominators.

Step 2: Simplify numerator and denominator.

$$ \begin{aligned} \frac{4a-1}{4} \cdot \frac{4a+1}{4} & \xlongequal{\text{Step 1}} \frac{ \left( 4a-1 \right) \cdot \left( 4a+1 \right) }{ 4 \cdot 4 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ 16a^2+ \cancel{4a} -\cancel{4a}-1 }{ 16 } = \frac{16a^2-1}{16} \end{aligned} $$