Factor $LaTeX: \displaystyle 12 x^{3} - 8 x^{2} - 48 x + 32$ .

Factoring out the GCF $LaTeX: \displaystyle 4$ from each term gives $LaTeX: \displaystyle 4(3 x^{3} - 2 x^{2} - 12 x + 8)$ . Grouping the first two terms and factoring out their GCF, $LaTeX: \displaystyle x^{2}$ , gives $LaTeX: \displaystyle x^{2}(3 x - 2)$ . Grouping the last two terms and factoring out their GCF, $LaTeX: \displaystyle -4$ , gives $LaTeX: \displaystyle -4(3 x - 2)$ . The polynomial now has a common binomial factor of $LaTeX: \displaystyle 3 x - 2$ . This gives $LaTeX: \displaystyle 4[x^{2} \left(3 x - 2\right) -4 \cdot \left(3 x - 2\right)] = 4\left(3 x - 2\right) \left(x^{2} - 4\right)$ . The quadratic factor can be factored using the difference of squares to give $LaTeX: \displaystyle 4\left(x - 2\right) \left(x + 2\right) \left(3 x - 2\right).$