Multiply $ \dfrac{3}{4} $ by $ x $ to get $ \dfrac{ 3x }{ 4 } $.

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.

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

Subtract $ \dfrac{3x}{4} $ from $ 2x^2 $ to get $ \dfrac{ \color{purple}{ 8x^2-3x } }{ 4 }$.

Step 1: Write $ 2x^2 $ 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 $\color{blue}{ 4 }$.

$$ \begin{aligned} 2x^2- \frac{3x}{4} & \xlongequal{\text{Step 1}} \frac{2x^2}{\color{red}{1}} - \frac{3x}{4} = \frac{ 2x^2 \cdot \color{blue}{ 4 }}{ 1 \cdot \color{blue}{ 4 }} - \frac{ 3x }{ 4 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 8x^2 } }{ 4 } - \frac{ \color{purple}{ 3x } }{ 4 }=\frac{ \color{purple}{ 8x^2-3x } }{ 4 } \end{aligned} $$

Add $ \dfrac{8x^2-3x}{4} $ and $ 3 $ to get $ \dfrac{ \color{purple}{ 8x^2-3x+12 } }{ 4 }$.

Step 1: Write $ 3 $ 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 $\color{blue}{4}$.

$$ \begin{aligned} \frac{8x^2-3x}{4} +3 & \xlongequal{\text{Step 1}} \frac{8x^2-3x}{4} + \frac{3}{\color{red}{1}} = \frac{ 8x^2-3x }{ 4 } + \frac{ 3 \cdot \color{blue}{ 4 }}{ 1 \cdot \color{blue}{ 4 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 8x^2-3x } }{ 4 } + \frac{ \color{purple}{ 12 } }{ 4 }=\frac{ \color{purple}{ 8x^2-3x+12 } }{ 4 } \end{aligned} $$

Multiply $x^2-4x+29$ by $ \dfrac{8x^2-3x+12}{4} $ to get $ \dfrac{8x^4-35x^3+256x^2-135x+348}{4} $.

Step 1: Write $ x^2-4x+29 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} x^2-4x+29 \cdot \frac{8x^2-3x+12}{4} & \xlongequal{\text{Step 1}} \frac{x^2-4x+29}{\color{red}{1}} \cdot \frac{8x^2-3x+12}{4} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( x^2-4x+29 \right) \cdot \left( 8x^2-3x+12 \right) }{ 1 \cdot 4 } \xlongequal{\text{Step 3}} \frac{ 8x^4-3x^3+12x^2-32x^3+12x^2-48x+232x^2-87x+348 }{ 4 } = \\[1ex] &= \frac{8x^4-35x^3+256x^2-135x+348}{4} \end{aligned} $$