1. **State the problem:** Factor the polynomial $$9x^3 - 54x^2 - x + 6$$ completely. 2. **Group terms:** Group the polynomial into two parts to factor by grouping: $$ (9x^3 - 54x^2) - (x - 6) $$ 3. **Factor out the greatest common factor (GCF) from each group:** $$ 9x^2(x - 6) - 1(x - 6) $$ 4. **Factor out the common binomial factor:** $$ (9x^2 - 1)(x - 6) $$ 5. **Recognize the difference of squares in the first factor:** $$ 9x^2 - 1 = (3x)^2 - 1^2 = (3x - 1)(3x + 1) $$ 6. **Write the fully factored form:** $$ (3x - 1)(3x + 1)(x - 6) $$ **Final answer:** $$ (3x - 1)(3x + 1)(x - 6) $$