$$ (3x+2)^4 = (3x+2)^2 \cdot (3x+2)^2 $$

Find $ \left(3x+2\right)^2 $ using formula.

$$ (A + B)^2 = \color{blue}{A^2} + 2 \cdot A \cdot B + \color{red}{B^2} $$

where $ A = \color{blue}{ 3x } $ and $ B = \color{red}{ 2 }$.

$$ \begin{aligned}\left(3x+2\right)^2 = \color{blue}{\left( 3x \right)^2} +2 \cdot 3x \cdot 2 + \color{red}{2^2} = 9x^2+12x+4\end{aligned} $$

Multiply each term of $ \left( \color{blue}{9x^2+12x+4}\right) $ by each term in $ \left( 9x^2+12x+4\right) $.

$$ \left( \color{blue}{9x^2+12x+4}\right) \cdot \left( 9x^2+12x+4\right) = 81x^4+108x^3+36x^2+108x^3+144x^2+48x+36x^2+48x+16 $$

Combine like terms:

$$ 81x^4+ \color{blue}{108x^3} + \color{red}{36x^2} + \color{blue}{108x^3} + \color{green}{144x^2} + \color{orange}{48x} + \color{green}{36x^2} + \color{orange}{48x} +16 = \\ = 81x^4+ \color{blue}{216x^3} + \color{green}{216x^2} + \color{orange}{96x} +16 $$

Multiply $ \color{blue}{2} $ by $ \left( x^3-1\right) $ $$ \color{blue}{2} \cdot \left( x^3-1\right) = 2x^3-2 $$

Multiply each term of $ \left( \color{blue}{2x^3-2}\right) $ by each term in $ \left( 6x-3\right) $.

$$ \left( \color{blue}{2x^3-2}\right) \cdot \left( 6x-3\right) = 12x^4-6x^3-12x+6 $$

Remove the parentheses by changing the sign of each term within them.

$$ - \left( 12x^4-6x^3-12x+6 \right) = -12x^4+6x^3+12x-6 $$

Combine like terms:

$$ \color{blue}{81x^4} + \color{red}{216x^3} +216x^2+ \color{green}{96x} + \color{orange}{16} \color{blue}{-12x^4} + \color{red}{6x^3} + \color{green}{12x} \color{orange}{-6} = \\ = \color{blue}{69x^4} + \color{red}{222x^3} +216x^2+ \color{green}{108x} + \color{orange}{10} $$