Multiply $4$ by $ \dfrac{x}{2} $ to get $ \dfrac{ 4x }{ 2 } $.

Step 1: Write $ 4 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 4 \cdot \frac{x}{2} & \xlongequal{\text{Step 1}} \frac{4}{\color{red}{1}} \cdot \frac{x}{2} \xlongequal{\text{Step 2}} \frac{ 4 \cdot x }{ 1 \cdot 2 } \xlongequal{\text{Step 3}} \frac{ 4x }{ 2 } \end{aligned} $$

Add $ \dfrac{6}{10x^2+10x} $ and $ \dfrac{4x}{2} $ to get $ \dfrac{ \color{purple}{ 40x^3+40x^2+12 } }{ 20x^2+20x }$.

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}{ 2 }$ and the second by $\color{blue}{ 10x^2+10x }$.

$$ \begin{aligned} \frac{6}{10x^2+10x} + \frac{4x}{2} & = \frac{ 6 \cdot \color{blue}{ 2 }}{ \left( 10x^2+10x \right) \cdot \color{blue}{ 2 }} + \frac{ 4x \cdot \color{blue}{ \left( 10x^2+10x \right) }}{ 2 \cdot \color{blue}{ \left( 10x^2+10x \right) }} = \\[1ex] &=\frac{ \color{purple}{ 12 } }{ 20x^2+20x } + \frac{ \color{purple}{ 40x^3+40x^2 } }{ 20x^2+20x }=\frac{ \color{purple}{ 40x^3+40x^2+12 } }{ 20x^2+20x } \end{aligned} $$