Multiply $ \dfrac{1}{10} $ by $ x $ to get $ \dfrac{ x }{ 10 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{1}{10} \cdot x & \xlongequal{\text{Step 1}} \frac{1}{10} \cdot \frac{x}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 1 \cdot x }{ 10 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ x }{ 10 } \end{aligned} $$

Multiply $ \dfrac{1}{5} $ by $ x^2 $ to get $ \dfrac{ x^2 }{ 5 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{1}{5} \cdot x^2 & \xlongequal{\text{Step 1}} \frac{1}{5} \cdot \frac{x^2}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 1 \cdot x^2 }{ 5 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ x^2 }{ 5 } \end{aligned} $$

Add $ \dfrac{x}{10} $ and $ \dfrac{x^2}{5} $ to get $ \dfrac{ \color{purple}{ 2x^2+x } }{ 10 }$.

To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{2}$.

$$ \begin{aligned} \frac{x}{10} + \frac{x^2}{5} & = \frac{ x }{ 10 } + \frac{ x^2 \cdot \color{blue}{ 2 }}{ 5 \cdot \color{blue}{ 2 }} = \\[1ex] &=\frac{ \color{purple}{ x } }{ 10 } + \frac{ \color{purple}{ 2x^2 } }{ 10 }=\frac{ \color{purple}{ 2x^2+x } }{ 10 } \end{aligned} $$