Factor $LaTeX: \displaystyle - 12 x^{3} - 54 x^{2} - 6 x - 27$ .

Factoring out the GCF $LaTeX: \displaystyle -3$ from each term gives $LaTeX: \displaystyle -3(4 x^{3} + 18 x^{2} + 2 x + 9)$ . Grouping the first two terms and factoring out their GCF, $LaTeX: \displaystyle 2 x^{2}$ , gives $LaTeX: \displaystyle 2 x^{2}(2 x + 9)$ . Grouping the last two terms and factoring out their GCF, $LaTeX: \displaystyle 1$ , gives $LaTeX: \displaystyle 1(2 x + 9)$ . The polynomial now has a common binomial factor of $LaTeX: \displaystyle 2 x + 9$ . This gives $LaTeX: \displaystyle -3[2 x^{2} \left(2 x + 9\right) +1 \cdot \left(2 x + 9\right)] = -3\left(2 x + 9\right) \left(2 x^{2} + 1\right)$ .