1. The problem is to simplify the expression: $$4x^2 - 6mn - 18mn^2$$ and verify the factorization $$-6mn(2 + 6n)$$. 2. First, observe the terms: $$4x^2$$, $$-6mn$$, and $$-18mn^2$$. 3. The factorization $$-6mn(2 + 6n)$$ applies only to the last two terms, not the entire expression. 4. Factor the last two terms: $$-6mn - 18mn^2 = -6mn(1 + 3n)$$ 5. Note that $$-6mn(2 + 6n)$$ is incorrect because: $$-6mn(2 + 6n) = -12mn - 36mn^2$$ which does not match the original terms. 6. Therefore, the correct factorization of the last two terms is: $$-6mn(1 + 3n)$$ 7. The full expression remains: $$4x^2 - 6mn - 18mn^2 = 4x^2 - 6mn(1 + 3n)$$ 8. This is the simplified and factored form. Final answer: $$4x^2 - 6mn(1 + 3n)$$