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

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

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

Divide $1$ by $ \dfrac{a+x}{a} $ to get $ \dfrac{ a }{ a+x } $.

Step 1: To divide rational expressions, multiply the first fraction by the reciprocal of the second fraction.

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

Step 3: Multiply numerators and denominators.

Step 4: Simplify numerator and denominator.

$$ \begin{aligned} \frac{1}{ \frac{\color{blue}{a+x}}{\color{blue}{a}} } & \xlongequal{\text{Step 1}} 1 \cdot \frac{\color{blue}{a}}{\color{blue}{a+x}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{1}{\color{red}{1}} \cdot \frac{a}{a+x} \xlongequal{\text{Step 3}} \frac{ 1 \cdot a }{ 1 \cdot \left( a+x \right) } \xlongequal{\text{Step 4}} \frac{ a }{ a+x } \end{aligned} $$

Multiply $ \dfrac{100}{2x^2+30x+100} $ by $ \dfrac{a}{a+x} $ to get $ \dfrac{ 100a }{ 2ax^2+2x^3+30ax+30x^2+100a+100x } $.

Step 1: Multiply numerators and denominators.

Step 2: Simplify numerator and denominator.

$$ \begin{aligned} \frac{100}{2x^2+30x+100} \cdot \frac{a}{a+x} & \xlongequal{\text{Step 1}} \frac{ 100 \cdot a }{ \left( 2x^2+30x+100 \right) \cdot \left( a+x \right) } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ 100a }{ 2ax^2+2x^3+30ax+30x^2+100a+100x } \end{aligned} $$