Add $6x+5$ and $ \dfrac{1}{x} $ to get $ \dfrac{ \color{purple}{ 6x^2+5x+1 } }{ x }$.

Step 1: Write $ 6x+5 $ 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 $\color{blue}{ x }$.

$$ \begin{aligned} 6x+5+ \frac{1}{x} & \xlongequal{\text{Step 1}} \frac{6x+5}{\color{red}{1}} + \frac{1}{x} = \frac{ \left( 6x+5 \right) \cdot \color{blue}{ x }}{ 1 \cdot \color{blue}{ x }} + \frac{ 1 }{ x } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 6x^2+5x } }{ x } + \frac{ \color{purple}{ 1 } }{ x }=\frac{ \color{purple}{ 6x^2+5x+1 } }{ x } \end{aligned} $$

Multiply $2x^2+3$ by $ \dfrac{6x^2+5x+1}{x} $ to get $ \dfrac{12x^4+10x^3+20x^2+15x+3}{x} $.

Step 1: Write $ 2x^2+3 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 2x^2+3 \cdot \frac{6x^2+5x+1}{x} & \xlongequal{\text{Step 1}} \frac{2x^2+3}{\color{red}{1}} \cdot \frac{6x^2+5x+1}{x} \xlongequal{\text{Step 2}} \frac{ \left( 2x^2+3 \right) \cdot \left( 6x^2+5x+1 \right) }{ 1 \cdot x } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 12x^4+10x^3+2x^2+18x^2+15x+3 }{ x } = \frac{12x^4+10x^3+20x^2+15x+3}{x} \end{aligned} $$