\[\begin{array}{l}
{x^4} - 6{x^3} + 27{x^2} - 54x + 32\\
= {x^4} - {x^3} - 5{x^3} + 5{x^2} + 22{x^2} - 22x - 32x + 32\\
= {x^3}\left( {x - 1} \right) - 5{x^2}\left( {x - 1} \right) + 22x\left( {x - 1} \right) - 32\left( {x - 1} \right)\\
= \left( {x - 1} \right)\left( {{x^3} - 5{x^2} + 22x - 32} \right)\\
= \left( {x - 1} \right)\left( {{x^3} - 2{x^2} - 3{x^2} + 6x + 16x - 32} \right)\\
= \left( {x - 1} \right)\left[ {{x^2}\left( {x - 2} \right) - 3x\left( {x - 2} \right) + 16\left( {x - 2} \right)} \right]\\
= \left( {x - 1} \right)\left( {x - 2} \right)\left( {{x^2} - 3x + 16} \right).
\end{array}\]
