1. **State the problem:** We have a polynomial $$p(n) = n^3 + 2n^2 - 23n + k$$ and we know that when divided by $$n + 6$$, the remainder is $$-1$$. We need to find the remainder when $$p(n)$$ is divided by $$n - 5$$. 2. **Use the Remainder Theorem:** The Remainder Theorem states that the remainder of dividing a polynomial $$p(n)$$ by $$n - a$$ is $$p(a)$$. 3. **Find $$k$$ using the first condition:** Since dividing by $$n + 6$$ is the same as dividing by $$n - (-6)$$, the remainder is $$p(-6) = -1$$. Calculate $$p(-6)$$: $$p(-6) = (-6)^3 + 2(-6)^2 - 23(-6) + k = -216 + 2(36) + 138 + k = -216 + 72 + 138 + k = (-216 + 72 + 138) + k = (-216 + 210) + k = -6 + k$$ Set equal to remainder: $$-6 + k = -1$$ Solve for $$k$$: $$k = -1 + 6 = 5$$ 4. **Find the remainder when dividing by $$n - 5$$:** By the Remainder Theorem, the remainder is $$p(5)$$. Calculate $$p(5)$$ with $$k=5$$: $$p(5) = 5^3 + 2(5)^2 - 23(5) + 5 = 125 + 2(25) - 115 + 5 = 125 + 50 - 115 + 5 = (125 + 50) - 115 + 5 = 175 - 115 + 5 = 60 + 5 = 65$$ **Final answer:** The remainder when dividing $$p(n)$$ by $$n - 5$$ is $$65$$.