1. We are asked to factor the quadratic expression $$15n^2 - 27n - 6$$. 2. The general form of a quadratic is $$ax^2 + bx + c$$. To factor, we look for two numbers that multiply to $$a \times c$$ and add to $$b$$. 3. Here, $$a = 15$$, $$b = -27$$, and $$c = -6$$. 4. Calculate $$a \times c = 15 \times (-6) = -90$$. 5. Find two numbers that multiply to $$-90$$ and add to $$-27$$. These numbers are $$-30$$ and $$3$$ because $$-30 \times 3 = -90$$ and $$-30 + 3 = -27$$. 6. Rewrite the middle term using these numbers: $$15n^2 - 30n + 3n - 6$$ 7. Group terms: $$(15n^2 - 30n) + (3n - 6)$$ 8. Factor each group: $$15n(n - 2) + 3(n - 2)$$ 9. Factor out the common binomial: $$(15n + 3)(n - 2)$$ 10. Factor out the common factor 3 from the first binomial: $$3(5n + 1)(n - 2)$$ 11. Final factored form is $$3(5n + 1)(n - 2)$$. Answer: $$15n^2 - 27n - 6 = 3(5n + 1)(n - 2)$$.