Factor $LaTeX: \displaystyle 54 x^{3} + 27 x^{2} - 18 x - 9$ .

Factoring out the GCF $LaTeX: \displaystyle 9$ from each term gives $LaTeX: \displaystyle 9(6 x^{3} + 3 x^{2} - 2 x - 1)$ . Grouping the first two terms and factoring out their GCF, $LaTeX: \displaystyle 3 x^{2}$ , gives $LaTeX: \displaystyle 3 x^{2}(2 x + 1)$ . Grouping the last two terms and factoring out their GCF, $LaTeX: \displaystyle -1$ , gives $LaTeX: \displaystyle -1(2 x + 1)$ . The polynomial now has a common binomial factor of $LaTeX: \displaystyle 2 x + 1$ . This gives $LaTeX: \displaystyle 9[3 x^{2} \left(2 x + 1\right) -1 \cdot \left(2 x + 1\right)] = 9\left(2 x + 1\right) \left(3 x^{2} - 1\right)$ .