Multiply $3x^2-8x+4$ by $ \dfrac{9x^2-3x-2}{9} $ to get $ \dfrac{27x^4-81x^3+54x^2+4x-8}{9} $.

Step 1: Write $ 3x^2-8x+4 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 3x^2-8x+4 \cdot \frac{9x^2-3x-2}{9} & \xlongequal{\text{Step 1}} \frac{3x^2-8x+4}{\color{red}{1}} \cdot \frac{9x^2-3x-2}{9} \xlongequal{\text{Step 2}} \frac{ \left( 3x^2-8x+4 \right) \cdot \left( 9x^2-3x-2 \right) }{ 1 \cdot 9 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 27x^4-9x^3-6x^2-72x^3+24x^2+16x+36x^2-12x-8 }{ 9 } = \frac{27x^4-81x^3+54x^2+4x-8}{9} \end{aligned} $$

Multiply $ \dfrac{27x^4-81x^3+54x^2+4x-8}{9} $ by $ x^2 $ to get $ \dfrac{ 27x^6-81x^5+54x^4+4x^3-8x^2 }{ 9 } $.

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.

$$ \begin{aligned} \frac{27x^4-81x^3+54x^2+4x-8}{9} \cdot x^2 & \xlongequal{\text{Step 1}} \frac{27x^4-81x^3+54x^2+4x-8}{9} \cdot \frac{x^2}{\color{red}{1}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( 27x^4-81x^3+54x^2+4x-8 \right) \cdot x^2 }{ 9 \cdot 1 } \xlongequal{\text{Step 3}} \frac{ 27x^6-81x^5+54x^4+4x^3-8x^2 }{ 9 } \end{aligned} $$

Subtract $4$ from $ \dfrac{27x^6-81x^5+54x^4+4x^3-8x^2}{9} $ to get $ \dfrac{ \color{purple}{ 27x^6-81x^5+54x^4+4x^3-8x^2-36 } }{ 9 }$.

Step 1: Write $ 4 $ 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 second fraction by $\color{blue}{9}$.

$$ \begin{aligned} \frac{27x^6-81x^5+54x^4+4x^3-8x^2}{9} -4 & \xlongequal{\text{Step 1}} \frac{27x^6-81x^5+54x^4+4x^3-8x^2}{9} - \frac{4}{\color{red}{1}} = \\[1ex] &= \frac{ 27x^6-81x^5+54x^4+4x^3-8x^2 }{ 9 } - \frac{ 4 \cdot \color{blue}{ 9 }}{ 1 \cdot \color{blue}{ 9 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 27x^6-81x^5+54x^4+4x^3-8x^2 } }{ 9 } - \frac{ \color{purple}{ 36 } }{ 9 }=\frac{ \color{purple}{ 27x^6-81x^5+54x^4+4x^3-8x^2-36 } }{ 9 } \end{aligned} $$

Multiply $ \dfrac{27x^6-81x^5+54x^4+4x^3-8x^2-36}{9} $ by $ 3x^2-5x-2 $ to get $ \dfrac{81x^8-378x^7+513x^6-96x^5-152x^4+32x^3-92x^2+180x+72}{9} $.

Step 1: Write $ 3x^2-5x-2 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{27x^6-81x^5+54x^4+4x^3-8x^2-36}{9} \cdot 3x^2-5x-2 & \xlongequal{\text{Step 1}} \frac{27x^6-81x^5+54x^4+4x^3-8x^2-36}{9} \cdot \frac{3x^2-5x-2}{\color{red}{1}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( 27x^6-81x^5+54x^4+4x^3-8x^2-36 \right) \cdot \left( 3x^2-5x-2 \right) }{ 9 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 81x^8-135x^7-54x^6-243x^7+405x^6+162x^5+162x^6-270x^5-108x^4+12x^5-20x^4-8x^3-24x^4+40x^3+16x^2-108x^2+180x+72 }{ 9 } = \\[1ex] &= \frac{81x^8-378x^7+513x^6-96x^5-152x^4+32x^3-92x^2+180x+72}{9} \end{aligned} $$